Patent Publication Number: US-2023142179-A1

Title: Systems And Methods For Controlling Movement Of A Surgical Tool Along A Predefined Path

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The subject application is a continuation of U.S. patent application Ser. No. 16/811,909, filed on Mar. 6, 2020, which claims priority to and all the benefits of U.S. Provisional Patent App. No. 62/815,739, filed Mar. 8, 2019, and U.S. Provisional Patent App. No. 62/896,394, filed Sep. 5, 2019, the contents of each of the above applications being hereby incorporated by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates generally to systems and methods for controlling movement of a surgical tool along a predefined path. 
     BACKGROUND 
     Robotic surgical systems perform surgical procedures at surgical sites. Robotic surgical systems typically include a manipulator and an end effector coupled to the manipulator. Often, the end effector comprises a surgical tool to remove tissue at the surgical site. 
     In a manual mode of operation, one type of robotic surgical system senses forces and torques manually applied to the surgical tool by a user. The robotic surgical system commands positioning of the surgical tool to emulate motion expected by the user from application of the sensed forces and torques. Thus, the robotic surgical system generally positions the surgical tool in accordance with the user&#39;s intentions and expectations so that the user, for example, is able to remove a desired volume of tissue. However, in the manual mode, it can be fatiguing for the user to cause movement of the surgical tool as needed to completely remove the entire volume of tissue, especially when the volume of tissue is relatively large compared to the size of the surgical tool. Accordingly, the robotic surgical system is also operable in a semi-autonomous mode in which the robotic surgical system commands the manipulator to move the surgical tool autonomously along a predefined tissue removal path, unassisted by the user. However, when operating in the semi-autonomous mode, there may be a perception that the user has less control over the surgical tool. For this reason, the manual mode may be preferred by some users. 
     There is a need in the art for systems and methods to address these challenges. 
     SUMMARY 
     A robotic surgical system is provided that comprises a surgical tool and a manipulator configured to support the surgical tool. The manipulator comprises a plurality of links. A force/torque sensor measures forces and torques applied to the surgical tool. A control system obtains a milling path for the surgical tool wherein the milling path is three-dimensional. The control system receives input from the force/torque sensor in response to a user manually applying user forces and torques to the surgical tool. The control system calculates a tangential component of force tangential to the milling path based on the input from the force/torque sensor and commands the manipulator to advance the surgical tool along the milling path based on the calculated tangential component. 
     A method is provided for operating a robotic surgical system. The robotic surgical system comprises a surgical tool, a manipulator configured to support the surgical tool, and a force/torque sensor to measure forces and torques applied to the surgical tool. The method comprises obtaining a milling path for the surgical tool wherein the milling path is three-dimensional. Input from the force/torque sensor is received in response to a user manually applying user forces and torques to the surgical tool. A tangential component of force tangential to the milling path is calculated based on the input from the force/torque sensor so that the manipulator can be commanded to advance the surgical tool along the milling path based on the calculated tangential component. 
     Another robotic surgical system is provided that comprises a surgical tool and a manipulator configured to support the surgical tool. The manipulator comprises a plurality of links. A force/torque sensor measures forces and torques applied to the surgical tool. A control system obtains a milling path for the surgical tool wherein the milling path is three-dimensional. The control system also defines virtual constraints on movement of the surgical tool along the milling path with respect to two degrees of freedom each being normal to the milling path. The virtual constraints are defined to constrain movement of the surgical tool to be along the milling path. The control system receives input from the force/torque sensor in response to a user manually applying user forces and torques to the surgical tool. The control system simulates dynamics of the surgical tool in a virtual simulation based on the virtual constraints and the input from the force/torque sensor and commands the manipulator to advance the surgical tool along the milling path based on the virtual simulation. 
     Another method is provided for operating a robotic surgical system. The robotic surgical system comprises a surgical tool, a manipulator configured to support the surgical tool, and a force/torque sensor to measure forces and torques applied to the surgical tool. The method comprises obtaining a milling path for the surgical tool wherein the milling path is three-dimensional. Virtual constraints are defined with respect to two degrees of freedom each being normal to the milling path. The virtual constrains are defined to constrain movement of the surgical tool to be along the milling path. Input from the force/torque sensor is received in response to a user manually applying user forces and torques to the surgical tool. Dynamics of the surgical tool are simulated in a virtual simulation based on the virtual constraints and the input from the force/torque sensor so that the manipulator can be commanded to advance the surgical tool along the milling path based on the virtual simulation. 
     Another robotic surgical system is provided that comprises a surgical tool and a manipulator configured to support the surgical tool. The manipulator comprises a plurality of links. A force/torque sensor measures forces and torques applied to the surgical tool. A control system obtains a milling path for the surgical tool wherein the milling path is three-dimensional. The control system receives an input from the force/torque sensor in response to user forces and torques manually applied to the surgical tool by a user. The control system calculates a tangential component of force tangential to the milling path based on the input from the force/torque sensor and calculates an effective feed rate for advancing the surgical tool along the milling path based on the calculated tangential component. The control system defines virtual constraints on movement of the surgical tool along the milling path with respect to three degrees of freedom and based on the effective feed rate to promote movement of the surgical tool along the milling path. The control system simulates dynamics of the surgical tool in a virtual simulation based on the virtual constraints and the input from the force/torque sensor and commands the manipulator to advance the surgical tool along the milling path based on the virtual simulation. 
     Another method is provided for operating a robotic surgical system. The robotic surgical system comprises a surgical tool, a manipulator configured to support the surgical tool, and a force/torque sensor to measure forces and torques applied to the surgical tool. The method comprises obtaining a milling path for the surgical tool wherein the milling path is three-dimensional. Input is received from the force/torque sensor in response to user forces and torques manually applied to the surgical tool by a user. A tangential component of force is calculated tangential to the milling path based on the input from the force/torque sensor. An effective feed rate is calculated based on the calculated tangential component. Virtual constraints are defined with respect to three degrees of freedom and based on the effective feed rate to promote movement of the surgical tool along the milling path. Dynamics of the surgical tool are simulated in a virtual simulation based on the virtual constraints and the input from the force/torque sensor so that the manipulator can be commanded to advance the surgical tool along the milling path based on the virtual simulation. 
     Another robotic surgical system is provided that comprises a surgical tool and a manipulator configured to support the surgical tool. The manipulator comprises a plurality of links. A force/torque sensor measures forces and torques applied to the surgical tool. A control system obtains a virtual boundary for the surgical tool wherein the virtual boundary is three-dimensional. The virtual boundary comprises a tube defining a milling path for the surgical tool. The control system defines virtual constraints on movement of the surgical tool inside the tube and along the milling path. The virtual constraints are defined to constrain movement of the surgical tool to be along the milling path. The control system receives an input from the force/torque sensor in response to user forces and torques manually applied to the surgical tool by a user. The control system simulates dynamics of the surgical tool in a virtual simulation based on the virtual constraints and the input from the force/torque sensor and commands the manipulator to advance the surgical tool along the milling path based on the virtual simulation. 
     Another method is provided for operating a robotic surgical system. The robotic surgical system comprises a surgical tool, a manipulator configured to support the surgical tool, and a force/torque sensor to measure forces and torques applied to the surgical tool. The method comprises obtaining a virtual boundary for the surgical tool wherein the virtual boundary is three-dimensional. The virtual boundary comprises a tube defining a milling path for the surgical tool. The method also comprises defining virtual constraints on movement of the surgical tool inside the tube and along the milling path, the virtual constraints being defined to constrain movement of the surgical tool to be along the milling path. Input is received from the force/torque sensor in response to user forces and torques manually applied to the surgical tool by a user. Dynamics of the surgical tool are simulated in a virtual simulation based on the virtual constraints and the input from the force/torque sensor and the manipulator is commanded to advance the surgical tool along the milling path based on the virtual simulation. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       Advantages of the present disclosure will be readily appreciated as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings. 
         FIG.  1    is a perspective view of a robotic surgical system. 
         FIG.  2    is a block diagram of a control system for controlling the robotic surgical system. 
         FIG.  3    is a functional block diagram of a software program. 
         FIG.  4    illustrates output of a boundary generator. 
         FIG.  5    illustrates output of a path generator. 
         FIGS.  6 A- 6 C  illustrate a sequence of steps carried out in a manual mode of operation of the robotic surgical system. 
         FIG.  7    illustrates a series of movements of the surgical tool along a milling path. 
         FIG.  8    is a block diagram of modules operable by the control system. 
         FIGS.  9 A- 9 J  illustrate each of the movements of the surgical tool along the milling path shown in  FIG.  7   . 
         FIG.  10    is a block diagram of modules operable by the control system. 
         FIG.  11    is an illustration of the milling path, showing the virtual constraints in x and y directions normal to the milling path, and a constraint application pose determined using a “nearest point” method. 
         FIG.  12    is an illustration of the milling path, showing the virtual constraints in x and y directions normal to the milling path, and a constraint application pose determined using a “virtual feed rate” method. 
         FIG.  13    is an illustration of a virtual rigid body and constraint parameters. 
         FIG.  14    shows a sample constraint equation. 
         FIGS.  15  and  16    show a sample forward dynamics algorithm for carrying out a virtual simulation. 
         FIG.  17    shows an example set of steps carried out by the control system to solve constraints, perform forward dynamics, and determine a commanded pose. 
         FIG.  18    illustrates a series of movements of the surgical tool along the milling path using the modules of  FIG.  10   . 
         FIGS.  19 A- 19 E  illustrate each of the movements of the surgical tool along the milling path shown in  FIG.  18   . 
         FIG.  20    is a block diagram of modules operable by the control system. 
         FIG.  21    is an illustration of the milling path, showing virtual constraints in x, y, and z directions with respect to a constraint coordinate system. 
         FIG.  22    illustrates a series of movements of the surgical tool along another milling path using the modules of  FIG.  20   . 
         FIGS.  23 A- 23 D  illustrate each of the movements of the surgical tool along the milling path shown in  FIG.  22   . 
         FIG.  24    illustrates another milling path and a virtual object defining a boundary to keep the surgical tool moving along the milling path, the virtual object being modeled as a triangulated mesh. 
         FIG.  25    illustrates another milling path and another virtual object defining a boundary to keep the surgical tool moving along the milling path. 
         FIG.  26    is a perspective view of a tele-manipulated robotic surgical system. 
     
    
    
     DETAILED DESCRIPTION 
     Referring to  FIG.  1   , a robotic surgical system  10  is illustrated. The system  10  is useful for treating a surgical site or anatomical volume (A) of a patient  12 , such as treating bone or soft tissue. In  FIG.  1   , the patient  12  is undergoing a surgical procedure. The anatomy in  FIG.  1    includes a femur F and a tibia T of the patient  12 . The surgical procedure may involve tissue removal or other forms of treatment. Treatment may include cutting, coagulating, lesioning the tissue, other in-situ tissue treatments, or the like. In some examples, the surgical procedure involves partial or total knee or hip replacement surgery, shoulder replacement surgery, spine surgery, or ankle surgery. In some examples, the system  10  is designed to cut away material to be replaced by surgical implants, such as hip and knee implants, including unicompartmental, bicompartmental, multicompartmental, or total knee implants. Some of these types of implants are shown in U.S. Patent Application Publication No. 2012/0330429, entitled, “Prosthetic Implant and Method of Implantation,” the disclosure of which is hereby incorporated by reference. The system  10  and techniques disclosed herein may be used to perform other procedures, surgical or non-surgical, or may be used in industrial applications or other applications where robotic systems are utilized. 
     The system  10  includes a manipulator  14 . The manipulator  14  has a base  16  and plurality of links  18 . A manipulator cart  17  supports the manipulator  14  such that the manipulator  14  is fixed to the manipulator cart  17 . The links  18  collectively form one or more arms of the manipulator  14 . The manipulator  14  may have a serial arm configuration (as shown in  FIG.  1   ), a parallel arm configuration, or any other suitable manipulator configuration. In other examples, more than one manipulator  14  may be utilized in a multiple arm configuration. 
     In the example shown in  FIG.  1   , the manipulator  14  comprises a plurality of joints J and a plurality of joint encoders  19  located at the joints J for determining position data of the joints J. For simplicity, only one joint encoder  19  is illustrated in  FIG.  1   , although other joint encoders  19  may be similarly illustrated. The manipulator  14  according to one example has six joints J 1 -J 6  implementing at least six-degrees of freedom (DOF) for the manipulator  14 . However, the manipulator  14  may have any number of degrees of freedom and may have any suitable number of joints J and may have redundant joints. 
     The manipulator  14  need not require joint encoders  19  but may alternatively, or additionally, utilize motor encoders present on motors at each joint J. Also, the manipulator  14  need not require rotary joints, but may alternatively, or additionally, utilize one or more prismatic joints. Any suitable combination of joint types are contemplated. 
     The base  16  of the manipulator  14  is generally a portion of the manipulator  14  that provides a fixed reference coordinate system for other components of the manipulator  14  or the system  10  in general. Generally, the origin of a manipulator coordinate system MNPL is defined at the fixed reference of the base  16 . The base  16  may be defined with respect to any suitable portion of the manipulator  14 , such as one or more of the links  18 . Alternatively, or additionally, the base  16  may be defined with respect to the manipulator cart  17 , such as where the manipulator  14  is physically attached to the manipulator cart  17 . In one example, the base  16  is defined at an intersection of the axes of joints J 1  and J 2 . Thus, although joints J 1  and J 2  are moving components in reality, the intersection of the axes of joints J 1  and J 2  is nevertheless a virtual fixed reference pose, which provides both a fixed position and orientation reference and which does not move relative to the manipulator  14  and/or manipulator cart  17 . In other examples, the manipulator  14  can be a hand-held manipulator where the base  16  is a base portion of a tool (e.g., a portion held free-hand by the user) and the tool tip is movable relative to the base portion. The base portion has a reference coordinate system that is tracked and the tool tip has a tool tip coordinate system that is computed relative to the reference coordinate system (e.g., via motor and/or joint encoders and forward kinematic calculations). Movement of the tool tip can be controlled to follow the path since its pose relative to the path can be determined. 
     The manipulator  14  and/or manipulator cart  17  house a manipulator controller  26 , or other type of control unit. The manipulator controller  26  may comprise one or more computers, or any other suitable form of controller that directs the motion of the manipulator  14 . The manipulator controller  26  may have a central processing unit (CPU) and/or other processors, memory (not shown), and storage (not shown). The manipulator controller  26  is loaded with software as described below. The processors could include one or more processors to control operation of the manipulator  14 . The processors can be any type of microprocessor, multi-processor, and/or multi-core processing system. The manipulator controller  26  may additionally, or alternatively, comprise one or more microcontrollers, field programmable gate arrays, systems on a chip, discrete circuitry, and/or other suitable hardware, software, or firmware that is capable of carrying out the functions described herein. The term processor is not intended to limit any embodiment to a single processor. The manipulator  14  may also comprise a user interface UI with one or more displays and/or input devices (e.g., push buttons, keyboard, mouse, microphone (voice-activation), gesture control devices, touchscreens, etc.). 
     A tool  20  couples to the manipulator  14  and is movable relative to the base  16  to interact with the anatomy in certain modes. The tool  20  is a physical and surgical tool and is or forms part of an end effector  22  supported by the manipulator  14  in certain embodiments. The tool  20  may be grasped by the user. One possible arrangement of the manipulator  14  and the tool  20  is described in U.S. Pat. No. 9,119,655, entitled, “Surgical Manipulator Capable of Controlling a Surgical Instrument in Multiple Modes,” the disclosure of which is hereby incorporated by reference. The manipulator  14  and the tool  20  may be arranged in alternative configurations. The tool  20  can be like that shown in U.S. Patent Application Publication No. 2014/0276949, filed on Mar. 15, 2014, entitled, “End Effector of a Surgical Robotic Manipulator,” hereby incorporated by reference. 
     The tool  20  includes an energy applicator  24  designed to contact and remove the tissue of the patient  12  at the surgical site. In one example, the energy applicator  24  is a bur  25 . The bur  25  may be substantially spherical and comprise a spherical center, radius (r) and diameter. Alternatively, the energy applicator  24  may be a drill bit, a saw blade, an ultrasonic vibrating tip, or the like. The tool  20  and/or energy applicator  24  may comprise any geometric feature, e.g., perimeter, circumference, radius, diameter, width, length, volume, area, surface/plane, range of motion envelope (along any one or more axes), etc. The geometric feature may be considered to determine how to locate the tool  20  relative to the tissue at the surgical site to perform the desired treatment. In some of the embodiments described herein, a spherical bur having a tool center point (TCP) will be described for convenience and ease of illustration, but is not intended to limit the tool  20  to any particular form. 
     The tool  20  may comprise a tool controller  21  to control operation of the tool  20 , such as to control power to the tool (e.g., to a rotary motor of the tool  20 ), control movement of the tool  20 , control irrigation/aspiration of the tool  20 , and/or the like. The tool controller  21  may be in communication with the manipulator controller  26  or other components. The tool  20  may also comprise a user interface UI with one or more displays and/or input devices (e.g., push buttons, keyboard, mouse, microphone (voice-activation), gesture control devices, touchscreens, etc.). The manipulator controller  26  controls a state (position and/or orientation) of the tool  20  (e.g, the TCP) with respect to a coordinate system, such as the manipulator coordinate system MNPL. The manipulator controller  26  can control (linear or angular) velocity, acceleration, or other derivatives of motion of the tool  20 . 
     The tool center point (TCP), in one example, is a predetermined reference point defined at the energy applicator  24 . The TCP has a known, or able to be calculated (i.e., not necessarily static), pose relative to other coordinate systems. The geometry of the energy applicator  24  is known in or defined relative to a TCP coordinate system. The TCP may be located at the spherical center of the bur  25  of the tool  20  such that only one point is tracked. The TCP may be defined in various ways depending on the configuration of the energy applicator  24 . The manipulator  14  could employ the joint/motor encoders, or any other non-encoder position sensing method, to enable a pose of the TCP to be determined. The manipulator  14  may use joint measurements to determine TCP pose and/or could employ techniques to measure TCP pose directly. The control of the tool  20  is not limited to a center point. For example, any suitable primitives, meshes, etc., can be used to represent the tool  20 . 
     The system  10  further includes a navigation system  32 . One example of the navigation system  32  is described in U.S. Pat. No. 9,008,757, filed on Sep. 24, 2013, entitled, “Navigation System Including Optical and Non-Optical Sensors,” hereby incorporated by reference. The navigation system  32  tracks movement of various objects. Such objects include, for example, the manipulator  14 , the tool  20  and the anatomy, e.g., femur F and tibia T. The navigation system  32  tracks these objects to gather state information of each object with respect to a (navigation) localizer coordinate system LCLZ. Coordinates in the localizer coordinate system LCLZ may be transformed to the manipulator coordinate system MNPL, and/or vice-versa, using transformations. 
     The navigation system  32  includes a cart assembly  34  that houses a navigation controller  36 , and/or other types of control units. A navigation user interface UI is in operative communication with the navigation controller  36 . The navigation user interface includes one or more displays  38 . The navigation system  32  is capable of displaying a graphical representation of the relative states of the tracked objects to the user using the one or more displays  38 . The navigation user interface UI further comprises one or more input devices to input information into the navigation controller  36  or otherwise to select/control certain aspects of the navigation controller  36 . Such input devices include interactive touchscreen displays. However, the input devices may include any one or more of push buttons, a keyboard, a mouse, a microphone (voice-activation), gesture control devices, and the like. 
     The navigation system  32  also includes a navigation localizer  44  coupled to the navigation controller  36 . In one example, the localizer  44  is an optical localizer and includes a camera unit  46 . The camera unit  46  has an outer casing  48  that houses one or more optical sensors  50 . The localizer  44  may comprise its own localizer controller  49  and may further comprise a video camera VC. 
     The navigation system  32  includes one or more trackers. In one example, the trackers include a pointer tracker PT, one or more manipulator trackers  52 A,  52 B, a first patient tracker  54 , and a second patient tracker  56 . In the illustrated example of  FIG.  1   , the manipulator tracker is firmly attached to the tool  20  (i.e., tracker  52 A), the first patient tracker  54  is firmly affixed to the femur F of the patient  12 , and the second patient tracker  56  is firmly affixed to the tibia T of the patient  12 . In this example, the patient trackers  54 ,  56  are firmly affixed to sections of bone. The pointer tracker PT is firmly affixed to a pointer P used for registering the anatomy to the localizer coordinate system LCLZ. The manipulator tracker  52 A,  52 B may be affixed to any suitable component of the manipulator  14 , in addition to, or other than the tool  20 , such as the base  16  (i.e., tracker  52 B), or any one or more links  18  of the manipulator  14 . The trackers  52 A,  52 B,  54 ,  56 , PT may be fixed to their respective components in any suitable manner. For example, the trackers may be rigidly fixed, flexibly connected (optical fiber), or not physically connected at all (ultrasound), as long as there is a suitable (supplemental) way to determine the relationship (measurement) of that respective tracker to the object that it is associated with. 
     Any one or more of the trackers may include active markers  58 . The active markers  58  may include light emitting diodes (LEDs). Alternatively, the trackers  52 A,  52 B,  54 ,  56 , PT may have passive markers, such as reflectors, which reflect light emitted from the camera unit  46 . Other suitable markers not specifically described herein may be utilized. 
     The localizer  44  tracks the trackers  52 A,  52 B,  54 ,  56 , PT to determine a state of each of the trackers  52 A,  52 B,  54 ,  56 , PT, which correspond respectively to the state of the object respectively attached thereto. The localizer  44  may perform known triangulation techniques to determine the states of the trackers  52 ,  54 ,  56 , PT, and associated objects. The localizer  44  provides the state of the trackers  52 A,  52 B,  54 ,  56 , PT to the navigation controller  36 . In one example, the navigation controller  36  determines and communicates the state the trackers  52 A,  52 B,  54 ,  56 , PT to the manipulator controller  26 . As used herein, the state of an object includes, but is not limited to, data that defines the position and/or orientation of the tracked object or equivalents/derivatives of the position and/or orientation. For example, the state may be a pose of the object, and may include linear velocity data, and/or angular velocity data, and the like. 
     The navigation controller  36  may comprise one or more computers, or any other suitable form of controller. Navigation controller  36  has a central processing unit (CPU) and/or other processors, memory (not shown), and storage (not shown). The processors can be any type of processor, microprocessor or multi-processor system. The navigation controller  36  is loaded with software. The software, for example, converts the signals received from the localizer  44  into data representative of the position and orientation of the objects being tracked. The navigation controller  36  may additionally, or alternatively, comprise one or more microcontrollers, field programmable gate arrays, systems on a chip, discrete circuitry, and/or other suitable hardware, software, or firmware that is capable of carrying out the functions described herein. The term processor is not intended to limit any embodiment to a single processor. 
     Although one example of the navigation system  32  is shown that employs triangulation techniques to determine object states, the navigation system  32  may have any other suitable configuration for tracking the manipulator  14 , tool  20 , and/or the patient  12 . In another example, the navigation system  32  and/or localizer  44  are ultrasound-based. For example, the navigation system  32  may comprise an ultrasound imaging device coupled to the navigation controller  36 . The ultrasound imaging device images any of the aforementioned objects, e.g., the manipulator  14 , the tool  20 , and/or the patient  12 , and generates state signals to the navigation controller  36  based on the ultrasound images. The ultrasound images may be 2-D, 3-D, or a combination of both. The navigation controller  36  may process the images in near real-time to determine states of the objects. The ultrasound imaging device may have any suitable configuration and may be different than the camera unit  46  as shown in  FIG.  1   . 
     In another example, the navigation system  32  and/or localizer  44  are radio frequency (RF)-based. For example, the navigation system  32  may comprise an RF transceiver coupled to the navigation controller  36 . The manipulator  14 , the tool  20 , and/or the patient  12  may comprise RF emitters or transponders attached thereto. The RF emitters or transponders may be passive or actively energized. The RF transceiver transmits an RF tracking signal and generates state signals to the navigation controller  36  based on RF signals received from the RF emitters. The navigation controller  36  may analyze the received RF signals to associate relative states thereto. The RF signals may be of any suitable frequency. The RF transceiver may be positioned at any suitable location to track the objects using RF signals effectively. Furthermore, the RF emitters or transponders may have any suitable structural configuration that may be much different than the trackers  52 A,  52 B,  54 ,  56 , PT shown in  FIG.  1   . 
     In yet another example, the navigation system  32  and/or localizer  44  are electromagnetically based. For example, the navigation system  32  may comprise an EM transceiver coupled to the navigation controller  36 . The manipulator  14 , the tool  20 , and/or the patient  12  may comprise EM components attached thereto, such as any suitable magnetic tracker, electro-magnetic tracker, inductive tracker, or the like. The trackers may be passive or actively energized. The EM transceiver generates an EM field and generates state signals to the navigation controller  36  based upon EM signals received from the trackers. The navigation controller  36  may analyze the received EM signals to associate relative states thereto. Again, such navigation system  32  examples may have structural configurations that are different than the navigation system  32  configuration shown in  FIG.  1   . 
     The navigation system  32  may have any other suitable components or structure not specifically recited herein. Furthermore, any of the techniques, methods, and/or components described above with respect to the navigation system  32  shown may be implemented or provided for any of the other examples of the navigation system  32  described herein. For example, the navigation system  32  may utilize solely inertial tracking or any combination of tracking techniques, and may additionally or alternatively comprise, fiber optic-based tracking, machine-vision tracking, and the like. 
     Referring to  FIG.  2   , the system  10  includes a control system  60  that comprises, among other components, the manipulator controller  26 , the navigation controller  36 , and the tool controller  21 . The control system  60  further includes one or more software programs and software modules shown in  FIG.  3   . The software modules may be part of the program or programs that operate on the manipulator controller  26 , navigation controller  36 , tool controller  21 , or any combination thereof, to process data to assist with control of the system  10 . The software programs and/or modules include computer readable instructions stored in non-transitory memory  64  on the manipulator controller  26 , navigation controller  36 , tool controller  21 , or a combination thereof, to be executed by one or more processors  70  of the controllers  21 ,  26 ,  36 . The memory  64  may be any suitable configuration of memory, such as RAM, non-volatile memory, etc., and may be implemented locally or from a remote database. Additionally, software modules for prompting and/or communicating with the user may form part of the program or programs and may include instructions stored in memory  64  on the manipulator controller  26 , navigation controller  36 , tool controller  21 , or any combination thereof. The user may interact with any of the input devices of the navigation user interface UI or other user interface UI to communicate with the software modules. The user interface software may run on a separate device from the manipulator controller  26 , navigation controller  36 , and/or tool controller  21 . 
     The control system  60  may comprise any suitable configuration of input, output, and processing devices suitable for carrying out the functions and methods described herein. The control system  60  may comprise the manipulator controller  26 , the navigation controller  36 , or the tool controller  21 , or any combination thereof, or may comprise only one of these controllers. These controllers may communicate via a wired bus or communication network as shown in  FIG.  2   , via wireless communication, or otherwise. The control system  60  may also be referred to as a controller. The control system  60  may comprise one or more microcontrollers, field programmable gate arrays, systems on a chip, discrete circuitry, sensors, displays, user interfaces, indicators, and/or other suitable hardware, software, or firmware that is capable of carrying out the functions described herein. 
     Referring to  FIG.  3   , the software employed by the control system  60  includes a boundary generator  66 . As shown in  FIG.  4   , the boundary generator  66  is a software program or module that generates a virtual boundary  71  for constraining movement and/or operation of the tool  20 . The virtual boundary  71  may be one-dimensional, two-dimensional, three-dimensional, and may comprise a point, line, axis, trajectory, plane, or other shapes, including complex geometric shapes. In some embodiments, the virtual boundary  71  is a surface defined by a triangle mesh. Such virtual boundaries  71  may also be referred to as virtual objects. The virtual boundaries  71  may be defined with respect to an anatomical model AM, such as a 3-D bone model. In the example of  FIG.  4   , the virtual boundaries  71  are planar boundaries to delineate five planes for a total knee implant, and are associated with a 3-D model of the head of the femur F. The anatomical model AM is registered to the one or more patient trackers  54 ,  56  such that the virtual boundaries  71  become associated with the anatomical model AM. The virtual boundaries  71  may be implant-specific, e.g., defined based on a size, shape, volume, etc. of an implant and/or patient-specific, e.g., defined based on the patient&#39;s anatomy. The virtual boundaries  71  may be boundaries that are created pre-operatively, intra-operatively, or combinations thereof. In other words, the virtual boundaries  71  may be defined before the surgical procedure begins, during the surgical procedure (including during tissue removal), or combinations thereof. In any case, the control system  60  obtains the virtual boundaries  71  by storing/retrieving the virtual boundaries  71  in/from memory, obtaining the virtual boundaries  71  from memory, creating the virtual boundaries  71  pre-operatively, creating the virtual boundaries  71  intra-operatively, or the like. 
     The manipulator controller  26  and/or the navigation controller  36  track the state of the tool  20  relative to the virtual boundaries  71 . In one example, the state of the TCP is measured relative to the virtual boundaries  71  for purposes of determining haptic forces to be applied to a virtual rigid body model via a virtual simulation so that the tool  20  remains in a desired positional relationship to the virtual boundaries  71  (e.g., not moved beyond them). The results of the virtual simulation are commanded to the manipulator  14 . The control system  60  controls/positions the manipulator  14  in a manner that emulates the way a physical handpiece would respond in the presence of physical boundaries/barriers. The boundary generator  66  may be implemented on the manipulator controller  26 . Alternatively, the boundary generator  66  may be implemented on other components, such as the navigation controller  36 . 
     Referring to  FIGS.  3  and  5   , a path generator  68  is another software program or module run by the control system  60 . In one example, the path generator  68  is run by the manipulator controller  26 . The path generator  68  generates a tool path TP for the tool  20  to traverse, such as for removing sections of the anatomy to receive an implant. The tool path TP may comprise a plurality of path segments PS, or may comprise a single path segment PS. The path segments PS may be straight segments, curved segments, combinations thereof, or the like. The tool path TP may also be defined with respect to the anatomical model AM. The tool path TP may be implant-specific, e.g., defined based on a size, shape, volume, etc. of an implant and/or patient-specific, e.g., defined based on the patient&#39;s anatomy. 
     In one version described herein, the tool path TP is defined as a tissue removal path, but, in other versions, the tool path TP may be used for treatment other than tissue removal. One example of the tissue removal path described herein comprises a milling path  72 . It should be understood that the term “milling path” generally refers to the path of the tool  20  in the vicinity of the target site for milling the anatomy and is not intended to require that the tool  20  be operably milling the anatomy throughout the entire duration of the path. For instance, as will be understood in further detail below, the milling path  72  may comprise sections or segments where the tool  20  transitions from one location to another without milling. Additionally, other forms of tissue removal along the milling path  72  may be employed, such as tissue ablation, and the like. The milling path  72  may be a predefined path that is created pre-operatively, intra-operatively, or combinations thereof. In other words, the milling path  72  may be defined before the surgical procedure begins, during the surgical procedure (including during tissue removal), or combinations thereof. In any case, the control system  60  obtains the milling path  72  by storing/retrieving the milling path  72  in/from memory, obtaining the milling path  72  from memory, creating the milling path  72  pre-operatively, creating the milling path  72  intra-operatively, or the like. The milling path  72  may have any suitable shape, or combinations of shapes, such as circular, helical/corkscrew, linear, curvilinear, combinations thereof, and the like. 
     One example of a system and method for generating the virtual boundaries  71  and/or the milling path  72  is described in U.S. Pat. No. 9,119,655, entitled, “Surgical Manipulator Capable of Controlling a Surgical Instrument in Multiple Modes,” the disclosure of which is hereby incorporated by reference. In some examples, the virtual boundaries  71  and/or milling paths  72  may be generated offline rather than on the manipulator controller  26  or navigation controller  36 . Thereafter, the virtual boundaries  71  and/or milling paths  72  may be utilized at runtime by the manipulator controller  26 . 
     Referring back to  FIG.  3   , two additional software programs or modules run on the manipulator controller  26  and/or the navigation controller  36 . One software module performs behavior control  74 . Behavior control  74  is the process of computing data that indicates the next commanded position and/or orientation (e.g., pose) for the tool  20 . In some cases, only the position of the TCP is output from the behavior control  74 , while in other cases, the position and orientation of the tool  20  is output. Output from the boundary generator  66 , the path generator  68 , and a force/torque sensor S may feed as inputs into the behavior control  74  to determine the next commanded position and/or orientation for the tool  20 . The behavior control  74  may process these inputs, along with one or more virtual constraints described further below, to determine the commanded pose. 
     The second software module performs motion control  76 . One aspect of motion control is the control of the manipulator  14 . The motion control  76  receives data defining the next commanded pose from the behavior control  74 . Based on these data, the motion control  76  determines the next position of the joint angles of the joints J of the manipulator  14  (e.g., via inverse kinematics and Jacobian calculators) so that the manipulator  14  is able to position the tool  20  as commanded by the behavior control  74 , e.g., at the commanded pose. In other words, the motion control  76  processes the commanded pose, which may be defined in Cartesian space, into joint angles of the manipulator  14 , so that the manipulator controller  26  can command the joint motors accordingly, to move the joints J of the manipulator  14  to commanded joint angles corresponding to the commanded pose of the tool  20 . In one version, the motion control  76  regulates the joint angle of each joint J and continually adjusts the torque that each joint motor outputs to, as closely as possible, ensure that the joint motor drives the associated joint J to the commanded joint angle. 
     The boundary generator  66 , path generator  68 , behavior control  74 , and motion control  76  may be sub-sets of a software program  78 . Alternatively, each may be software programs that operate separately and/or independently in any combination thereof. The term “software program” is used herein to describe the computer-executable instructions that are configured to carry out the various capabilities of the technical solutions described. For simplicity, the term “software program” is intended to encompass, at least, any one or more of the boundary generator  66 , path generator  68 , behavior control  74 , and/or motion control  76 . The software program  78  can be implemented on the manipulator controller  26 , navigation controller  36 , or any combination thereof, or may be implemented in any suitable manner by the control system  60 . 
     A clinical application  80  may be provided to handle user interaction. The clinical application  80  handles many aspects of user interaction and coordinates the surgical workflow, including pre-operative planning, implant placement, registration, bone preparation visualization, and post-operative evaluation of implant fit, etc. The clinical application  80  is configured to output to the displays  38 . The clinical application  80  may run on its own separate processor or may run alongside the navigation controller  36 . In one example, the clinical application  80  interfaces with the boundary generator  66  and/or path generator  68  after implant placement is set by the user, and then sends the virtual boundary  71  and/or tool path TP returned by the boundary generator  66  and/or path generator  68  to the manipulator controller  26  for execution. Manipulator controller  26  executes the tool path TP as described herein. The manipulator controller  26  may additionally create certain segments (e.g., lead-in segments) when starting or resuming machining to smoothly get back to the generated tool path TP. The manipulator controller  26  may also process the virtual boundaries  71  to generate corresponding virtual constraints as described further below. 
     The system  10  may operate in a manual mode, such as described in U.S. Pat. No. 9,119,655, incorporated herein by reference. Here, the user manually directs, and the manipulator  14  executes movement of the tool  20  and its energy applicator  24  at the surgical site. The user physically contacts the tool  20  to cause movement of the tool  20  in the manual mode. In one version, the manipulator  14  monitors forces and torques placed on the tool  20  by the user in order to position the tool  20 . For example, the manipulator  14  may comprise the force/torque sensor S that detects the forces and torques applied by the user and generates corresponding input used by the control system  60  (e.g., one or more corresponding input/output signals). 
     The force/torque sensor S may comprise a 6-DOF force/torque transducer. The manipulator controller  26  and/or the navigation controller  36  receives the input (e.g., signals) from the force/torque sensor S. In response to the user-applied forces and torques, the manipulator  14  moves the tool  20  in a manner that emulates the movement that would have occurred based on the forces and torques applied by the user. Movement of the tool  20  in the manual mode may also be constrained in relation to the virtual boundaries  71  generated by the boundary generator  66 . In some versions, measurements taken by the force/torque sensor S are transformed from a force/torque coordinate system FT of the force/torque sensor S to another coordinate system, such as a virtual mass coordinate system VM in which a virtual simulation is carried out on the virtual rigid body model of the tool  20  so that the forces and torques can be virtually applied to the virtual rigid body in the virtual simulation to ultimately determine how those forces and torques (among other inputs) would affect movement of the virtual rigid body, as described below. 
     The system  10  may also operate in a semi-autonomous mode in which the manipulator  14  moves the tool  20  along the milling path  72  (e.g., the active joints J of the manipulator  14  operate to move the tool  20  without requiring force/torque on the tool  20  from the user). An example of operation in the semi-autonomous mode is also described in U.S. Pat. No. 9,119,655, incorporated herein by reference. In some embodiments, when the manipulator  14  operates in the semi-autonomous mode, the manipulator  14  is capable of moving the tool  20  free of user assistance. Free of user assistance may mean that a user does not physically contact the tool  20  to move the tool  20 . Instead, the user may use some form of remote control to control starting and stopping of movement. For example, the user may hold down a button of the remote control to start movement of the tool  20  and release the button to stop movement of the tool  20 . 
     In the manual mode, it may be fatiguing to the user to remove an entire volume of tissue Vt required to be removed for a particular surgical procedure, especially when the volume of tissue Vt is relatively large as compared to the working end of the tool  20 . As shown in  FIGS.  6 A through  6 C , for example, it may be difficult for the user, through manual mode operation of the manipulator  14 , to place the TCP of the tool  20  to remove all of the bone required. Instead, as illustrated in  FIG.  6 B , the user&#39;s operation in the manual mode (see tortuous movement arrow) causes the bur  25  to skip several subvolumes Vs of bone that need removal leaving these bone subvolumes Vs for later removal, as shown in  FIG.  6 C . To this end, the system  10  may switch from the manual mode to the semi-autonomous mode to complete the removal of the bone, such as in the manner described in U.S. Pat. No. 9,119,655, incorporated herein by reference, including generating a milling path  72  through these subvolumes Vs. Accordingly, to finish bone removal in preparation for receiving the implant, the manipulator  14  autonomously moves the TCP along the milling path  72 . 
     The system  10  may also operate in a guided-manual mode to remove the remaining subvolumes Vs of bone, or for other purposes. In this mode, aspects of control used in both the manual mode and the semi-autonomous mode are utilized. For example, forces and torques applied by the user are detected by the force/torque sensor S to determine an external force F ext . The external force F ext  may comprise other forces and torques, aside from those applied by the user, such as gravity-compensating forces, backdrive forces, and the like, as described in U.S. Pat. No. 9,119,655, incorporated herein by reference. Thus, the user-applied forces and torques at least partially define the external force F ext , and in some cases, may fully define the external force F ext . Additionally, in the guided-manual mode, the system  10  utilizes the milling path  72  (or other tool path) generated by the path generator  68  to help guide movement of the tool  20  along the milling path  72 . In some cases, the milling path  72  is generated upon the user entering the guided-manual mode. The guided-manual mode relies on manual manipulation of the tool  20  to advance the tool  20 , but such advancement, instead of merely emulating the movement that would have occurred based on the forces and torques applied by the user, is actively controlled to be along the milling path  72 . Therefore, the guided-manual mode combines direct user engagement with the tool  20  and the benefits associated with moving the tool  20  along the milling path  72 , as illustrated in  FIG.  7   . 
     In some cases, as shown in  FIG.  7   , a lead-in path  72   a  may first be generated by the behavior control  74  when initiating operation in the guided-manual mode. The lead-in path  72  is a direct pathway from the current position/pose of the tool  20  to the milling path  72  (e.g., connects the TCP to a starting point SP of the milling path  72 ). The lead-in path  72  may be generated based on a shortest distance from the current position/pose of the tool  20  to the milling path  72  (e.g., shortest distance from the TCP to any point on the milling path  72 ) or may be generated from the current position/pose of the tool  20  to the starting point SP of the milling path  72 . In any case, the lead-in path  72   a  is generated to lead the tool  20  to the milling path  72 . Creation of the lead-in path  72   a  may require the TCP of the tool  20 , and/or any other part of the tool  20 , to be within a predefined distance of the starting point SP. Visualization on the display  38  may guide the user into moving the tool  20  to be within this predefined distance. Such a distance could be defined by a virtual sphere  73  centered on the starting point SP, or the like, as shown in  FIG.  7   . Once within the predefined distance, the behavior control  74  creates the lead-in path  72   a  to guide the tool  20  to the remainder of the milling path  72 . This lead-in path  72   a  and switching to the guided-manual mode could be automatic upon the TCP of the tool  20  entering the sphere  73 . 
       FIG.  8    shows processes carried out to execute the guided-manual mode, in one example. In this example, the behavior control  74  comprises a path handler  82 . The path handler  82  comprises executable software stored in a non-transitory memory of any one or more of the aforementioned controllers and implemented by the control system  60 . As shown, one input into the path handler  82  comprises the milling path  72  generated by the path generator  68 . The milling path  72  may be three-dimensional, as previously described, and defined with respect to any desired coordinate system, such as the manipulator coordinate system MNPL, localizer coordinate system LCLZ, or other coordinate system. 
     Another input into the path handler  82 , in the example shown in  FIG.  8   , is the current external force F ext , including the forces/torques sensed by the force/torque sensor S. The external force F ext  may comprise three components of force along x, y, z axes, or may comprise six components of force including the three components of force along x, y, z axes and three components of torque about the x, y, z axes. The components of force may be initially defined in a sensor coordinate system, but can be transformed to another coordinate system, such as a coordinate system of the tool  20 . 
     Another input into the path handler  82  comprises the last (most recent or current) commanded pose CP. As previously mentioned, the commanded pose CP may comprise Cartesian coordinates of the TCP of the tool  20  and a commanded orientation of the tool  20 , e.g., pose. 
     The path handler  82  interpolates along the milling path  72  to determine the next commanded pose CP based on the external force F ext  (as transformed to the TCP) and the previous commanded pose. At each iteration of the process shown in  FIG.  8   , which may be carried out at any suitable frame rate (e.g., every 125 microseconds), the path handler  82  calculates a tangential component F ext  (tan) of the external force F ext , which is tangential to the milling path  72  at the previous commanded pose. This tangential component of force F ext  (tan) at least partially dictates how far along the milling path  72  the tool  20  should move. Said differently, since this tangential component of force F ext  (tan) is at least partially derived by how much force the user applied in the tangential direction (e.g., to move the tool  20  in the tangential direction), it largely (and completely, in some cases) defines how far the path handler  82  will ultimately decide to move the tool  20  along the milling path  72 . F ext  (tan) is computed using both the force and torque components measured by the force/torque sensor S in the sensor coordinate system, which are then transformed to the TCP (tangential path direction) using a Jacobian. 
     In some cases, the user may exert a force on the tool  20  indicating a desire to move off the milling path  72  and end operation in the guided-manual mode, such as by applying a force normal to the milling path  72  and/or opposite to a previous force applied to move the tool  20  along the milling path  72 . For example, the sign+/− of the tangential component of the external force F ext  (tan) and/or the magnitude of the normal component of F ext  may be evaluated by the control system  60 . Based on the sign+/− of the tangential component of F ext , and/or the magnitude of the normal component of F ext  exceeding a predetermined threshold, the control system  60  can determine whether to switch back to the manual mode or to provide some other type of control for the user, i.e., in the event the user&#39;s application of force clearly indicates a desire to end the guided-manual mode. In some versions, the sign+/− of the tangential component of the external force F ext  (tan) is utilized to determine a direction that the user wishes to move along the milling path  72  (e.g., forward/backward), and not necessarily a desire to exit the guided-manual mode. 
     Once the tangential component F ext  (tan) of the external force F ext  is determined and has a sign+/− indicating a desire to continue moving along the milling path  72  and/or the magnitude of the normal component indicates a desire to continue moving along the milling path  72 , the path handler  82  generates the next commanded pose CP. Referring to  FIGS.  8  and  9 A , the next commanded pose can be determined, in one example, by the following steps: (1) selecting a virtual mass (Vm) for the tool  20 , which may be close to the actual mass of the tool  20 , or may be any appropriate value to provide desired behavior; (2) calculating a virtual acceleration (a) for the TCP of the tool  20  in the tangential direction based on the tangential component of force F ext  (tan) applied at the TCP and the virtual mass (F ext (tan)=(Vm)*(a)); (3) integrating the acceleration over one time step (e.g., 125 microseconds) to determine velocity; (4) integrating the velocity over the same time step to determine a change in position and corresponding translation distance in the tangential direction; and (5) applying the translation distance from the previous commanded pose, along the milling path  72 , to yield the next commanded pose CP, e.g., the next commanded pose CP is located on the milling path  72  spaced from the previous commanded pose by the translation distance. The computed translation distance may thus be used as an approximation of how far to move the TCP of the tool  20  along the milling path  72 . 
     The scalar virtual mass Vm could be chosen to give a desired feel, i.e., how the TCP of the tool  20  should accelerate along the milling path  72  in response to the user&#39;s tangential forces. Alternately, the scalar virtual mass Vm could be computed from a 6-DOF virtual mass matrix M, which contains the mass and inertia matrix for the virtual rigid body, by computing the scalar effective mass each time step that is seen along the tangential direction of the milling path  72 . 
     The commanded pose CP may also comprise an orientation component that may be defined separately from determining how far to move the TCP of the tool  20  along the milling path  72 . For example, the orientation component may be based on maintaining a constant orientation while the TCP moves along the milling path  72 . The orientation component may be a predefined orientation or defined based on a particular location along the milling path  72  being traversed. The orientation component may also be user-defined and/or defined in other ways. Thus, the commanded pose CP may include the position output computed by the path handler  82  combined with the separately-computed orientation. The position output and the orientation are combined and output as the commanded pose CP to the motion control  76 . 
     The path handler  82  generates the commanded poses CP to be located on the milling path  72  such that the TCP of the tool  20  is generally constrained to movement along the milling path  72 . The milling path  72  may be non-linear between consecutive commanded poses CP, so in some iterations the TCP may move off the milling path  72  temporarily, but is generally constrained to return to the milling path  72  by virtue of the commanded poses CP being located on the milling path  72 . The description herein of the tool  20  being constrained to movement along the milling path  72  does not require the tool  20  to be restricted to movement only on the milling path  72  but refers to movement generally along the milling path  72  as shown in  FIG.  18   , for example. 
     Once the next commanded pose CP is determined, the motion control  76  commands the manipulator  14  to advance the TCP of the tool  20  along the milling path  72  to the next commanded pose CP. Thus, in this version, the control system  60  is configured to command the manipulator  14  to advance the tool  20  along the milling path  72  based on the calculated tangential component of force F ext  (tan) and to the exclusion of components of the external force F ext  normal to the milling path  72 . 
       FIGS.  9 A to  9 J  illustrate a number of iterations of the process of  FIG.  8   , starting from an initial time (t 0 ) to a final time (t 9 ) in which the tool  20  traverses the entire milling path  72 , moving from a first commanded pose CP 1  (see  FIG.  9 B ) to a final commanded pose CP 9  (see  FIG.  9 J ). Initially, to enter the guided-manual mode, the user may provide some form of input into the control system  60 , such as via the user interface UT on the tool  20  and/or the manipulator  14 , or the guided-manual mode may be selectable via the clinical application  80 . The guided-manual mode may also be triggered by moving the tool  20  towards the starting point SP and/or by being within a predefined distance of the starting point SP. One or more of the controllers  21 ,  26 ,  36  receive this input and initiate the process shown in  FIG.  8   . In a first step, at the initial time (t 0 ), the control system  60  generates the lead-in path  72   a , as shown in  FIG.  9 A . Once the lead-in path  72   a  is generated and the TCP is thereby located on the milling path  72 , then the external force F ext  at the initial time (t 0 ) can be measured/calculated, applied to the TCP, and resolved into forces normal to the milling path  72  (F ext  (n)) and forces tangential to the milling path  72  (F ext  (tan)). The next commanded pose of the TCP can then be determined as explained above. The motion control  76  then operates the manipulator  14  to move the TCP to the next commanded pose CP on the milling path  72  (compare  FIG.  9 A  to  FIG.  9 B ,  FIG.  9 B  to  FIG.  9 C , and so on). This process continues through to the final time (t 9 ) in  FIG.  9 J . At each iteration, the control system  60  may constrain the orientation of the tool  20  to remain in the same orientation as shown, or may control orientation in other ways, as described above. 
     At each step shown in  FIGS.  9 A through  9 J , the control system  60  may monitor progress of the tool  20  and virtually remove from the milling path  72  the segments of the milling path  72  traversed by the tool  20 . As a result, if the user switches to the semi-autonomous mode after operating for a period of time in the guided-manual mode, then the control system  60  will control the tool  20  to move along only the remaining segments of the milling path  72  not yet traversed while the tool  20  was operating in the guided-manual mode. The control system  60  may also define a new starting point SP for the milling path  72  based on the remaining segments. In other versions, the segments may not be removed so that the milling path  72  remains intact while in the guided-manual mode. 
       FIG.  10    illustrates processes carried out to execute the guided-manual mode in another example. In this example, the behavior control  74  comprises the path handler  82 , and additionally comprises a path constraint calculator  84 , a constraint solver  86 , and a virtual simulator  88 . The behavior control  74  further comprises a boundary handler  89  to generate boundary constraints based on the one or more virtual boundaries  71  generated by the boundary generator  66 . In some versions, there are no virtual boundaries  71  being used to constrain movement of the tool  20 , and so there would be no boundary constraints employed. The path handler  82 , path constraint calculator  84 , constraint solver  86 , virtual simulator  88 , and boundary handler  89  each comprise executable software stored in a non-transitory memory of any one or more of the aforementioned controllers and implemented by the control system  60 . 
     In this version, the path handler  82  receives two inputs: the milling path  72  and a previous commanded pose CP. The path handler  82  outputs a constraint application pose AP. The constraint application pose AP is located on the milling path  72 , as shown in  FIG.  11   . The path constraint calculator  84  computes two virtual constraints normal to the milling path  72  as defined by the constraint application pose AP. The constraint solver  86  calculates a constraint force F c  to be virtually applied to the tool  20  in the virtual simulator  88  based on the two virtual constraints where the two virtual constraints act to effectively cancel out components of user forces at the TCP normal to the milling path  72  (which may otherwise cause the TCP of the tool  20  to move normal to the milling path  72 ), thereby constraining movement of the TCP of the tool  20  to remain along the milling path  72 . The virtual simulator  88  may take into account the effects of other forces/constraints as well, as described further below. Ultimately, the virtual simulator  88  calculates a next commanded pose based on its virtual simulation, which ideally causes movement of the tool  20  along the milling path  72 . Thus, the user is able to manually manipulate the tool  20 , while the control system  60  assists in guiding the tool movement to be along the milling path  72 , by utilizing the two virtual constraints. The control system  60  thus assists with guiding the tool movement along the milling path  72  by not defining or at least limiting any constraints to movement of the tool  20  with respect to one degree of freedom being tangential to the milling path  72 . The net result is that user forces and torques applied to the tool  20  act to slide the TCP of the tool  20  along the milling path  72 . Non-constrained components of the user forces and torques are free to influence the overall movement of the tool  20 . Non-constrained forces may include the tangential component of user force at the TCP, as described, but may also include forces/torques that may act to reorient the tool  20 . Because there are no constraints in those directions, the virtual simulator  88  does not inhibit movement of the virtual rigid body in those directions. 
     The constraint application pose AP can be determined/defined using one or more methods. One method is referred to herein as the “nearest point” method and another method is referred to herein as the “virtual feed rate” method. Referring to  FIGS.  10  and  11   , in the “nearest point” method, the path handler  82  determines each constraint application pose AP by projecting the previous commanded pose CP onto the nearest point on the milling path  72 . In most cases, the nearest point on the milling path  72  is defined by a normal line NL extending from the previous commanded pose CP to the milling path  72 . Thus, the constraint application pose AP is defined by projecting the previous commanded pose along the normal line NL to the milling path  72 . Ideally, the origin of the constraint application pose AP is coincident with the origin of the previous commanded pose if the TCP of the tool  20  is being controlled to be exactly centered on the milling path  72 . However, due to the characteristics of the constraints being used (e.g., having configurable stiffness and damping), due to the existence of other constraints, due to the shape of the milling path  72 , and/or due to other factors, the previous commanded pose CP may not be perfectly centered on the milling path  72  (see the commanded poses CP 1 -CP 4  in  FIG.  18   , for example). Thus, the constraint application pose AP is calculated based on the spatial relationship of the previous commanded pose in the utilized coordinate system, relative to the milling path  72 , which can also be expressed in the same coordinate system. 
     In the “nearest point” method, a search process is employed to determine which of the path segments of the milling path  72  is the closest to the previous commanded pose CP. In one embodiment, the search process includes the path handler  82  first performing a broad-phase search to determine a subset of path segments PS within a specified distance of the previous commanded pose CP (e.g., within 1-10 mm, which could be bounded by how far the tool  20  could move relative to the anatomy in one time step at maximum relative velocity). Then, for each of these path segments PS returned from the broad-phase search, the path handler  82  performs a narrow-band search to compute the normal line NL and its length from the previous commanded pose CP to each of these path segments PS. The path handler  82  then selects the normal line NL that has the shortest length. This is the path segment PS to use, i.e., the closest path segment PS to the previous commanded pose CP. Where that normal line NL intersects that path segment PS determines the position of the constraint application pose AP along the milling path  72 . This process is repeated each time step as the commanded pose CP is updated. 
     The orientation component of the constraint application pose AP gives the orientation of the milling path  72  at that point. One possible convention for encoding this orientation information is to choose the z-axis of the constraint application pose coordinate system to be along the milling path  72  (pointing in a positive direction tangentially along the milling path  72 ) at that point, and the x- and y-axes pointing normal to the milling path  72  at that point (see  FIG.  11   ). Other conventions are possible, as long as the normal and tangent directions of the milling path  72  are captured. This path orientation information is employed by the path constraint calculator  84  to compute the virtual constraints, which with the earlier described convention would be in the x- and y-directions defined by the constraint application pose AP. 
     The constraint application pose AP can be expressed with respect to a suitable coordinate system, such as the manipulator coordinate system MNPL or localizer coordinate system LCLZ. If the path handler  82  is outputting the constraint application pose AP with respect to the manipulator coordinate system MNPL, then the path handler  82  first employs an appropriate transform computed elsewhere by the control system  60 , so that it can convert the milling path pose as well, such as from an anatomy tracker coordinate system to the manipulator coordinate system MNPL (i.e., the milling path  72  may be originally defined with respect to a respective anatomy tracker coordinate frame). 
     In the “virtual feed rate” method, instead of finding the nearest point on the milling path  72 , the path handler  82  computes a distance traveled from a previous commanded pose to a current commanded pose CP (see  FIG.  12   ). The path handler  82  then applies this distance along each path segment of the milling path  72  starting from a previous constraint application pose AP (from previous time step) to compute a location of a new constraint application pose AP. More specifically, the path handler  82  steps iteratively within a time step along the milling path  72  one path segment PS at a time until the accumulated distance stepped along the milling path  72  (e.g., path distance) is equal to the computed distance traveled from the previous commanded pose to the current commanded pose CP. This may include the path handler  82  repeatedly checking if the next segment&#39;s distance would exceed the computed distance, and if so, the path handler  82  then interpolating linearly within that path segment PS to determine the precise location along the path segment PS where the computed distance is reached. This location becomes the origin of the next constraint application pose AP. This iterative path interpolation process may also include smoothing filters on the interpolated path points, either time domain or spatial, acceleration filters, etc., before setting the result as the origin of the next constraint application pose AP. As with the “nearest point” method, the orientation component of the constraint application pose AP gives the orientation of the milling path  72  at the new constraint application pose AP and can be set as previously described. 
     Referring back to  FIG.  10   , the calculated constraint application pose AP is output from the path handler  82  and input into the path constraint calculator  84 . The path constraint calculator  84  outputs two virtual constraints (x constraint, y constraint) normal to the milling path  72  at the origin of the constraint application pose AP, as shown in  FIG.  11   . The path constraint calculator  84  can determine the normal directions using the orientation information of the constraint application pose AP, i.e., its x and y directions, such as when employing the convention for defining the orientation of the constraint application pose AP described above. 
     As previously described, the two virtual constraints are provided to effectively cancel out components of user forces at the TCP normal to the milling path  72 . Without canceling such normal components of user forces, the TCP of the tool  20  could otherwise move away from the milling path  72 . The two virtual constraints thereby act to constrain movement of the TCP of the tool  20  so that the TCP remains generally on the milling path  72 . The virtual constraints are defined so that virtual forces can be calculated and applied in the virtual simulation to cancel out the components of user forces at the TCP normal to the milling path  72 , and hold the user on the milling path  72  with suitable accuracy. In some versions, the two virtual constraints are velocity impulse constraints. In some versions, the constraints are similar to those used in the impulse modeling described in U.S. Pat. No. 9,119,655, incorporated herein by reference. In some versions, these virtual constraints are defined exclusively in the guided-manual mode, and not in the manual mode or the semi-autonomous mode. In some versions, virtual constraints are used in all modes. 
     As previously noted, the control system  60  may not provide any constraints to movement of the tool  20  with respect to one degree of freedom being tangential to the milling path  72 . For example, the path constraint calculator  84  may define the virtual constraints in only two degrees of freedom, normal to the milling path  72 , but not define or provide any constraints with respect to the degree of freedom tangential to the milling path  72 . As described further below, however, other constraints employed in the behavior control  74  may affect movement of the tool  20  tangential to the milling path  72 , such as damping constraints, boundary constraints, etc. For example, in some cases, one or more virtual constraints could be defined in the tangential direction along the milling path  72  to provide damping. These damping constraints could be tuned as desired to control how easy or difficult movement along the milling path  72  in the guided-manual mode feels to the user. Less damping makes it easier to move the tool. More damping is sometimes desired to provide increased stability while machining. 
     Referring to  FIG.  13   , the virtual constraints are defined primarily by three runtime parameters: a constraint Jacobian Jp, which maps each one-dimensional, virtual constraint between a constraint coordinate system and a coordinate system employed for the virtual simulation (e.g., between the constraint application pose coordinate system and the virtual mass coordinate system VM); a desired velocity Vp 2 , which is a scalar velocity of the constraint in the constraint coordinate system (e.g., the desired velocity may be zero when the patient is immobile and the associated milling path  72  defined relative to the patient is not moving, but may be other than zero when the patient moves since the milling path  72  is tied to the patient); and a constraint distance Δd, which is how close the TCP is to the constraint and dictates whether the constraint is being violated. In this case, Δd refers to a distance from the milling path  72 , and the two virtual constraints are violated any time the distance is greater than zero in the x and y directions. 
     In some embodiments, the virtual constraints are not perfectly rigid, but instead each of the virtual constraints has tuning parameters to adjust the stiffness of the virtual constraints, e.g., by incorporating spring and damping parameters into the constraints. 
     Each virtual constraint also has configuration settings. The configuration settings may comprise: information regarding tuning parameters, such as a constraint force mixing parameter (C) and an error reduction parameter (c); upper and/or lower force limits; and/or upper and lower constraint distance offsets. The upper and lower force limits refer to limits on the forces computed for each constraint that are ultimately solved by the constraint solver  86  to produce a constraint force F c , as described further below. Since the two virtual constraints being provided by the path constraint calculator  84  are two-sided constraints (e.g., the constraint forces computed to satisfy the constraints can be positive or negative), the force limits can be set high in positive and negative directions (e.g., −100,000/+100,000 Newtons) or at any desired limit. The upper and lower constraint distance offsets dictate when the constraint is active. With respect to the two virtual constraints described above, the upper and lower constraint distance offsets can be set so that the constraint is active any time the TCP is off the milling path  72  (e.g., at 0 mm). 
     Referring to  FIG.  14   , the constraint parameters for each virtual constraint are passed into the constraint solver  86  by the path constraint calculator  84 . The constraint solver  86  places the constraint data for each constraint into a corresponding row of a constraint equation, in matrix form, to solve for F p . The constraint data is placed in the constraint equation, along with other information known by the constraint solver  86 , such as the external force F cgext , a damping force F damping , an inertial force F inertial , the virtual mass matrix M, a virtual mass velocity V cg1 , and the time step Δt (e.g., 125 microseconds). 
     The virtual mass matrix M combines 3×3 mass and inertia matrices. The damping and inertial forces F damping  and F inertial  are calculated/known by the virtual simulator  88  and are based on the virtual mass velocity V cg1  (e.g., the velocity of the virtual mass coordinate system VM) output by the virtual simulator  88  in a prior time step. The virtual mass velocity V cg1  is a 6-DOF velocity vector comprising linear and angular velocity components. The damping force F damping  is a 6-DOF force/torque vector computed as a function of the virtual mass velocity V cg1  and a damping coefficient matrix (linear and rotational coefficients may not be equal). Damping is applied to the virtual mass Vm to improve its stability. The inertial force F inertial  is also a 6-DOF force/torque vector computed as a function of the virtual mass velocity V cg1  and the virtual mass matrix M. The damping and inertial forces, F damping  and F inertial , can be determined in the manner described in U.S. Pat. No. 9,566,122 to Bowling et al., hereby incorporated herein by reference. 
     Referring to the constraint equation shown in  FIG.  14   , F p  is a force vector in the constraint coordinate system, i.e., each component of F p  is a scalar constraint force acting in the corresponding constraint direction. In order to solve for F p , as describe below, the equation shown in  FIG.  14    is converted into a matrix equation where each row represents a single, one-dimensional constraint. 
     The constraint solver  86  may be configured with any suitable algorithmic instructions (e.g., an iterative constraint solver, Projected Gauss-Seidel solver, etc.) to solve this system of constraint equations in order to provide a solution best satisfying the system of equations (e.g., best satisfying the various constraints). In some cases, all constraints may not simultaneously be met. For example, in the case where motion is overconstrained by the various constraints, the constraint solver  86  will essentially find a ‘best fit’ solution given the relative stiffness/damping of the various constraints. The constraint solver  86  solves the system of equations and ultimately outputs the constraint force F c . 
     When a Projected Gauss-Seidel solver is employed, the constraint solver  86  constructs A and b matrices based on the constraints, uses Projected Gauss-Seidel to solve the system of equations to determine the resulting force vector F p , takes the output of Projected Gauss-Seidel and transforms it from the constraint coordinate system to the virtual mass coordinate system VM. For example, using the equation F c =J p   T  F p , wherein F c  is the constraint force, each resulting force vector F p  is converted to a force/torque vector applied to the virtual mass coordinate system VM. 
     Methods of using Projected Gauss-Seidel to solve a system of equations for multiple constraints is shown, for example, in “Constraint based physics solver” by Marijn Tamis and Giuseppe Maggiore, dated Jun. 15, 2015 (v1.02), which can be found at http://www.mft-spirit.nl/files/MTamis_ConstraintBasedPhysicsSolver.pdf, or in “Comparison between Projected Gauss-Seidel and Sequential Impulse Solvers for Real-Time Physics Simulations,” by Marijn Tamis, dated Jul. 1, 2015 (v1.01), which can be found at http://www.mft-spirit.nl/files/MTamis_PGS_SI_Comparison.pdf, both of which are hereby incorporated herein by reference in their entirety. 
     The Projected Gauss-Seidel method addresses Linear Complementarity Problems (LCP). Inequality associated with LCP arises since some constraint types (e.g., one-sided constraints, such as the boundary constraints) can only push (apply force) in one direction, e.g., positive constraint force. If the calculated force for such a constraint is negative (or, more broadly, outside its allowed range) for a given iteration of the constraint solver  86 , which is invalid, the given constraint must be pruned (or alternately limited/capped at its upper or lower allowed value) and the remaining constraints solved, until a suitable result (i.e., convergence) is found. In this manner, the constraint solver  86  determines the active set of constraints for a given time step, and then solves for their values. Other constraint types can apply forces in both positive and negative directions, e.g., two-sided constraints. Such constraints include the x and y virtual constraints used to hold the tool  20  on the milling path  72  to limit movement in the normal direction to the milling path  72 . Such two-sided constraints, when enabled, are usually active and not pruned/limited during the constraint solver  86  iterations. 
     The constraint force F c  calculated by the constraint solver  86  comprises three components of force along x, y, z axes and three components of torque about the x, y, z axes. The virtual simulator  88  utilizes the constraint force F c , along with the external force F cgext , the damping force F damping , and the inertial force F inertial  (all of which may comprise six components of force/torque), in its virtual simulation. In some cases, these components of force/torque are first transformed into a common coordinate system (e.g., the virtual mass coordinate system VM) and then summed to define a total force F T . The resulting 6-DOF force (i.e., force and torque) is applied to the virtual rigid body and the resulting motion is calculated by the virtual simulator  88 . The virtual simulator  88  thus acts to effectively simulate how the various constraints, among other things, affects motion of the virtual rigid body. The virtual simulator  88  performs forward dynamics to calculate the resulting 6-DOF pose and velocity of the virtual rigid body based on the given total force F T  being applied to the virtual rigid body. In one example, the virtual simulator  88  comprises a physics engine, which is executable software stored in a non-transitory memory of any one or more of the aforementioned controllers  21 ,  26 ,  36  and implemented by the control system  60 . 
     For the virtual simulation, the virtual simulator  88  models the tool  20  as the virtual rigid body in the virtual mass coordinate system VM with the origin of the virtual mass coordinate system VM being located at the center of mass of the virtual rigid body, and with the coordinate axes being aligned with the principal axes of the virtual rigid body (see, e.g.,  FIG.  13   ). The virtual rigid body is a dynamic object and a rigid body representation of the tool  20  for purposes of the virtual simulation. The virtual rigid body is free to move according to six degrees of freedom (6-DOF) in Cartesian space according to the virtual simulation. The virtual simulation may be processed computationally without visual or graphical representations. Thus, it is not required that the virtual simulation display dynamics of the virtual rigid body. In other words, the virtual rigid body need not be modeled within a graphics application executed on a processing unit. The virtual rigid body may exist only for the virtual simulation. 
     The virtual rigid body and its properties (mass, inertia matrix, center of mass, principal axes, etc.) define how the tool  20  will move in response to applied forces and torques (applied by the user via the force/torque sensor S and based on the action of calculated constraint forces). It governs whether the tool  20  will feel heavy or light and how it will move (e.g., accelerate in translation and rotation) in response to applied forces and torques. By adjusting the properties of the virtual rigid body, the control system  60  can adjust how the tool  20  feels to the user. It may be desirable to have the properties of the virtual rigid body modeled to be reasonably close to the actual properties of the tool  20 , for as realistic motion/feel as possible, but that is not required. For control stability reasons (given the finite acceleration of the robot system, control latencies, etc.), the virtual mass and inertia may be modeled to be somewhat higher than that of the tool  20 . 
     The virtual rigid body may correspond to components, which may be on or within the tool  20 . Additionally or alternatively, the virtual rigid body may extend, in part, beyond the tool  20 . The virtual rigid body may take into account the tool  20  with the energy applicator  24  or may take into account the tool  20  without the energy applicator  24 . Furthermore, the virtual rigid body may be based on the TCP. In one example, the center of mass of the virtual rigid body is understood to be the point around which the virtual rigid body would rotate if a virtual force is applied to another point of the virtual rigid body and the virtual rigid body were otherwise unconstrained, i.e., not constrained by the manipulator  14 . The center of mass of the virtual rigid body may be close to, but need not be the same as, the actual center of mass of the tool  20 . The center of mass of the virtual rigid body can be determined empirically. Once the tool  20  is attached to the manipulator  14 , the position of the center of mass can be reset to accommodate the preferences of the individual practitioners. 
     The virtual simulator  88  effectively simulates rigid body dynamics of the tool  20  by virtually applying forces and/or torques on the virtual rigid body, i.e., by virtually applying the components of force and torque from the total force F T  on the center of mass of the virtual rigid body in the virtual mass coordinate system VM. Thus, the forces/torques virtually applied to the virtual rigid body may comprise forces/torques associated with the external force F cgext , the damping force F damping , the inertial force F inertial , and the forces/torques from the constraint force F c  associated with the various constraints (by virtue of being embodied in the constraint force F c ). 
     Rigid body Jacobians can be used to transform velocities and forces from one coordinate system (reference frame) to another on the same virtual rigid body and may be employed here to transform the forces and torques of F ext  to the virtual mass coordinate system VM as well (e.g., to yield F cgext  used in the constraint equation). The virtual simulator  88  then internally calculates the damping force F damping  and the inertial force F inertial  to determine the total force F T , and also to output the damping force F damping  and the inertial force F inertial  for use by the constraint solver  86  in its system of equations in the next time step. 
     A virtual forward dynamics algorithm, as shown in  FIGS.  16  and  17   , may be employed in the virtual simulation to simulate the motion of the virtual rigid body as it would move upon application of the total force F T . Effectively, the virtual forward dynamics algorithm solves the equation F=ma (or a=F/M) in 6-DOF and integrates the acceleration to yield velocity, which is then used to determine a new pose, as shown in  FIG.  17   . The control system  60  inputs the virtual forces and/or torques (e.g., the total force F T ) into the virtual simulator  88  and these virtual forces and/or torques are applied to the virtual rigid body at the center of mass (e.g., the CG) in the virtual simulation  88  when the virtual rigid body is in the initial pose with the initial velocity. The virtual rigid body is moved to a final pose having a different state (i.e., position and/or orientation) and with a final velocity within Cartesian space in response to the control system  60  satisfying the inputted virtual forces and/or torques. The next commanded pose CP to be sent to the motion control  76  is based on the final pose calculated by the virtual simulator  88 . Thus, the virtual simulator  88  operates to determine the next commanded pose CP by simulating the effects of applying the total force F T  on the virtual rigid body using virtual forward dynamics as shown in  FIG.  16   . 
     Velocity limits may be imposed on the virtual rigid body in the simulation. In some cases, the velocity limits may be set high so that they generally don&#39;t affect the simulation, or they may be set at any desired value. The virtual rigid body is in an initial pose (initial state) and has an initial velocity at commencement of each iteration of the virtual simulation (e.g., at each time step/interval dt). The initial pose and initial velocity may be defined as the final pose and the final velocity output by the virtual simulator  88  in the previous time step. 
       FIG.  17    summarizes various steps carried out by the behavior control  74 . These include steps performed by the constraint solver  86  and the virtual simulator  88  as described above. In step  100 , the external force F ext  is calculated based on readings taken from the force/torque sensor S. In step  102 , the constraints data associated with the various virtual constraints are fed into the constraint solver  86  from the path constraint calculator  84 , from the boundary handler  89 , and/or from other constraint sources. 
     In steps  104 - 108 , rigid body calculations are carried out by the virtual simulator  88  to determine the inverse mass matrix M −1 , the inertial force F inertial , and the damping force F damping  of the virtual rigid body. In steps  110 - 114 , the constraint solver  86  utilizes the output from the rigid body calculations performed in steps  104 - 108  and the constraints data provided in step  102  to perform the constraint force calculations previously described to ultimately yield the constraint force F c . In step  116 , the constraint force F c  is summed with the external force F cgext , the damping force F damping , and the inertial force F inertial  to yield the total force F T . In step  118 , the total force F T  is applied to the virtual rigid body in the virtual simulation conducted by the virtual simulator  88  to determine a new pose and velocity of the virtual rigid body in step  120 , and ultimately to transform the new pose and velocity to the TCP in step  122 . The new commanded pose CP (T TCP ), and velocity (V TCP ) are output by the virtual simulator  88  in step  124 . 
       FIG.  18    illustrates motion of the TCP of the tool  20  along the milling path  72  in the guided-manual mode using the process of  FIG.  10   . The TCP of the tool  20  moves from the starting position SP to a first commanded pose CP 1 , then to a second commanded pose CP 2 , then to a third commanded pose CP 3 , and finally to a fourth commanded pose CP 4 . Along the way, the control system  60  executes the process set forth in  FIG.  10    to determine the series of constraint application poses AP 0 , AP 1 , AP 2 , AP 3 , so that the control system  60  is able to determine each subsequent commanded pose CP. The purpose of each of the constraint application poses AP 0 , AP 1 , AP 2 , AP 3 , as described above, is to provide a reference pose at which to define the two virtual constraints so that the TCP remains generally on the milling path  72  at each iteration. For example, after the TCP moves to the first commanded pose CP 1 , the path handler  82  then calculates the first constraint application pose AP 1  (e.g., with x- and y-axes normal to the milling path  72  using the convention previously described), and the path constraint calculator  84  thereafter calculates the two normal, virtual constraints to be applied to the TCP of the tool  20  so that the next commanded pose CP 2  of the TCP of the tool  20  is along the milling path  72 . 
       FIGS.  19 A through  19 E  show the process being carried out from an initial time (t 0 ) to a final time (t 4 ). It should be noted that the application of user forces in the tangential direction along the milling path  72  causes the TCP of the tool  20  to slide generally along the milling path  72 , but in the absence of any external force F ext , the TCP would simply move from the previous commanded pose CP toward the next constraint application point AP. For example, referring to  FIG.  19 B  and  FIG.  19 C , if there was no external force F ext  applied as shown in  FIG.  19 B , then the TCP would have been pulled towards the constraint application pose AP 1  shown in  FIG.  19 C  (e.g., the next commanded pose CP 2  would be near or at the constraint application pose AP 1 , as opposed to being further along the milling path  72  as shown). This resulting movement is due to F ext . Any off-path components of F ext  are effectively canceled out by the constraint force F c , and the remaining (unconstrained) force components along the milling path  72  are allowed to slide the tool  20 . At each time step, the tool  20  can slide along the tangent line (defined by the z direction of the constraint application pose AP), and since the milling path  72  may be curved and the virtual constraints not infinitely stiff, the tool  20  may deviate slightly from the milling path  72  each time step. However, this deviation is compensated for the next time step by the action of the continually updating (i.e., at each time step) normal virtual constraints and resulting constraint force F c . 
     In some versions, the control system  60  operates to address behavior of the tool  20  at the start and/or end of the milling path  72  so that the user does not inadvertently over-machine and/or “slip off” the ends of the milling path  72 . To provide a rigid boundary at the ends of the milling path  72 , additional constraints, such as additional virtual constraints output by the path constraint calculator  84  may feed into the constraint solver  86 . For example, an end constraint may be added that is applied in the tangential direction (i.e., along the milling path  72 ), with the constraint applied in a direction which opposes movement that would result in moving past the start/end of path. Where the two virtual constraints implemented normal to the path direction are “two-sided” constraints (meaning they can yield both positive/negative forces as needed), the end constraint is a “one-sided” constraint (which can yield force only in the “positive” tangential direction as needed). The end constraint is defined tangentially along the milling path  72 . 
     The path handler  82  generates/sets an indicator (e.g., a flag or flags, such as ‘start path’ or ‘end path’ flags) that is sent to the path constraint calculator  84  along with the constraint application pose AP to trigger the path constraint calculator  84  to output the one-sided constraint in the tangential direction and oppose movement in the tangential direction such that it prevents further movement along the milling path  72 , but only in the direction that would result in the user progressing past the end of the milling path  72 . The direction of the end constraint may be based on the constraint application pose AP (e.g., applied opposite to its positive z-axis direction when approaching the end of path or applied in the positive z-axis direction when at the start of path). Other sign conventions to define the end constraint are also possible. For example, the constraint application pose AP may be set so that its positive z-axis direction is always pointing tangentially along the milling path  72  in a direction towards interior segments of the milling path  72  (i.e., the positive z-axis would be flipped once the midway point of the milling path  72  is reached). In this case, the end constraint would be applied in the positive tangential direction. 
     When the software detects the constraint application pose AP is at one of the end-of-path locations, it will enable the end constraint defined above; otherwise the end constraint is disabled. This approach allows the user to guide the tool  20  smoothly forward and backwards along the milling path  72 , but to have a firm stop (i.e., additional force feedback, with configurable stiffness/damping) at the corresponding ends of the milling path  72 . If a firm end-of-path behavior is not desired (i.e., if it&#39;s desired for the user to “fall off” the ends), an alternate approach would be to use the previously defined two virtual constraints normal to the milling path  72 , and simply turn them both off when an end-of-path target is reached (as determined by the path handler  82 ), allowing the tool  20  to move freely. This behavior could be accompanied by user feedback on the displays  38 , audible feedback, and/or haptic feedback. 
       FIG.  20    illustrates processes carried out to execute the guided-manual mode in another example. In this example, the behavior control  74  comprises the path handler  82 , the path constraint calculator  84 , the constraint solver  86 , and the virtual simulator  88 . The path handler  82 , path constraint calculator  84 , constraint solver  86 , and the virtual simulator  88  each comprises executable software stored in a non-transitory memory of any one or more of the aforementioned controllers and implemented by the control system  60 . 
     In this version, the path handler  82  receives three inputs: the milling path  72 , the previous commanded pose CP, and the external force F ext . The path handler  82  processes these inputs to determine a constraint application pose, at which to define three virtual constraints (x constraint, y constraint, z constraint), as illustrated in  FIG.  21   . In this version, the constraint application pose is a target pose TP with its origin located on the milling path  72  to which it is desired to move the TCP of the tool  20 . The same convention for locating the axes of the constraint application pose on the milling path  72  described above could be employed in this version as well to locate the axes of the target pose TP on the milling path  72 . Other conventions are also possible. Ideally, the next commanded pose CP coincides with the target pose TP. Of course, in certain cases, the next commanded pose CP may not coincide with the target pose TP. The external force F ext  is utilized by the path handler  82  to determine a virtual feed rate of the TCP of the tool  20 , which is then used by the path handler  82  to determine the next target pose TP. The origin of the next target pose TP is also located on the milling path  72 . 
     Additional constraints could also be used beyond the x, y, z constraints. For example, additional constraints could be used so that the target pose is also encoded with a desired orientation in one or more of the rotational degrees of freedom. In that case, one or more of the axes for the target pose TP could be chosen to give the desired orientation of the TCP coordinate frame (for that point on the milling path  72 ). Accordingly, more than three virtual constraints would be computed by the path constraint calculator  84  for both the position and orientation components. Thus, the guided-manual mode may assist users in guiding the TCP of the tool  20  along the milling path  72 , while also guiding the orientation of the tool  20  in one or more degrees of freedom. Path-defined orientations could be computed/determined by the path generator  68 , either offline or during the procedure, based on the surgical approach, clinical access, etc., and passed into the path handler  82  as part of the milling path  72 , such that the orientation of the tool  20  automatically changes in a desirable/predefined way as the user slides the TCP of the tool  20  along the milling path  72 . Alternatively, or additionally, a set of orientation constraints could be determined independently of the path constraint calculator  84  and passed into the constraint solver  86  as part of the ‘other constraints’ input. One example for this approach would be to have a 2-DOF set of orientation constraints (e.g., for a spherical bur) to keep the bur shaft within a predefined virtual aperture, as described in U.S. Pat. No. 9,566,122, hereby incorporated herein by reference. Other options for orientation control are possible, such as no orientation control whereby the user is able to freely reorient the tool  20 . These examples of orientation control could be employed in this version, the versions of  FIG.  8  or  10   , or in other embodiments. Thus, two or three virtual constraints for the TCP position could come from the path constraint calculator  84  (see, e.g.,  FIGS.  10  and  20   ), and additional constraints could be provided by an independent orientation control source. The position and orientation constraints, or individual constraints, can have different stiffness/damping tuning to give a desired user interaction and feel. The constraint solver  86  solves the full set of constraints and outputs a commanded pose CP to the motion control  76 . 
     At each iteration of the process shown in  FIG.  20   , which may be carried out at any suitable frame rate (e.g., every 125 microseconds), the path handler  82  calculates a tangential component F ext  (tan) of the external force F ext  that is tangential to the milling path  72  at the origin of the previous commanded pose CP. This tangential component of force F ext (tan) at least partially dictates how far along the milling path  72  the tool  20  should move. Said differently, since this tangential component of force F ext  (tan) is at least partially derived by how much force the user applied in the tangential direction (e.g., to move the tool  20  in the tangential direction), it largely defines how far to move the tool  20  along the milling path  72 . 
     Once the tangential component F ext  (tan) of the external force F ext  is determined, according to some examples, the tangential component F ext  (tan) is fed into the equation F ext  (tan)=(Vm)*(a) referenced above with respect to  FIG.  8    to define an effective feed rate along the milling path  72 , e.g., by integrating the acceleration over the time step (e.g., 125 microseconds), and to determine how far along the milling path  72  to place the origin of the target pose TP by integrating the effective feed rate (velocity) over the same time step. It should be appreciated that the effective feed rate (velocity) could be combined with other feed rate sources before determining the location of the target pose TP, such as the feed rate sources described in U.S. Pat. No. 9,566,122, hereby incorporated herein by reference. 
     F ext  (tan) can be computed so that a positive value is in a forward path direction and a negative value is in a negative path direction. The integrations can thus be performed in a manner yielding the direction of the tangential distance. For example, a negative distance indicates to move backwards along the milling path  72 . In some cases, it may not be desired to allow the user to move backwards along the milling path  72 . In that case, if the tangential distance computed is negative, it may be limited at zero. 
     Once the tangential distance and direction is determined (see  FIG.  22   , for example), the path handler  82  then steps iteratively within a time step along the milling path  72 , one path segment PS at a time, until the accumulated distance stepped along the milling path  72  (e.g., path distance) is equal to the tangential distance. This iterative process may include the path handler  82  repeatedly checking if the next segment&#39;s distance would exceed the computed tangential distance, and if so, the path handler  82  then interpolates linearly within that path segment PS to determine the precise location along the path segment PS where the tangential distance is reached. This location becomes the origin of the next target pose TP.  FIG.  23 B , for example, illustrates the tangential distance being greater than a distance of one full path segment 1PS, but shorter than the distance of two path segments 1PS, 2PS. Accordingly, the path handler  82  would linearly interpolate along the second path segment 2PS to determine the precise location along the second path segment 2PS where the tangential distance is reached and this becomes the next target pose TP. It may also help to imagine the tangential distance being folded over onto the second path segment 2PS as illustrated by the arrow in  FIG.  23 B . This iterative path interpolation process may also include smoothing filters on the interpolated path points, either time domain or spatial, acceleration filters, etc., before setting the result as the origin of the next target pose TP. The target pose TP is then sent to the path constraint calculator  84  to compute the three virtual constraints. In this example, these virtual constraints include x, y, z virtual constraints to be applied to effectively move the tool  20  from the current commanded pose CP of the tool  20  to the target pose TP (more or less constraints are also possible). These three constraints are computed based on the difference between the current commanded pose and the target pose TP. Of course, in other versions, orientation constraints could also be defined based on differences between current orientations and desired orientations. 
     The three virtual constraints defined by the path constraint calculator  84  are next input into the constraint solver  86  (possibly with boundary constraints and/or other constraints, as shown in  FIG.  20   ) to be processed by the constraint solver  86  to determine the resulting constraint force F c  in the same manner as previously described. The virtual simulator  88  then carries out a virtual simulation in the same manner previously described to ultimately determine a next commanded pose CP. 
       FIG.  22    illustrates motion of the TCP of the tool  20  along the milling path  72  in the guided-manual mode for this example from the starting point SP to a first commanded pose CP 1 , then to a second commanded pose CP 2 , and finally to a third commanded pose CP 3 . Along the way, the control system  60  executes the process set forth in  FIG.  20    to determine the series of target poses TP 1 , TP 2 , TP 3 , so that the control system  60  is able to determine each subsequent commanded pose CP in the virtual simulation. Again, although the commanded poses CP 1 , CP 2 , CP 3  are shown to coincide with the target poses TP 1 , TP 2 , TP 3 , this may not always be the case. 
     It should be appreciated that a lead-in path could be employed in the manner described above to guide the user to the starting point SP. From the starting point SP, upon the user manually applying forces and torques to the tool  20 , the external force F ext  is determined via the force/torque sensor S and the path handler  82  receives the external force F ext  to compute the effective feed rate and associated tangential distance to be traversed along the milling path  72  to define the first target pose TP 1 . Once the first target pose TP 1  is defined, the path constraint calculator  84  then calculates the three virtual constraints to ultimately cause movement of the tool  20  to/toward the target pose TP 1 . The constraint solver  86  and virtual simulator  88  operate as before to determine each subsequent commanded pose. 
       FIGS.  23 A through  23 D  show the process being carried out from an initial time (t 0 ) to a final time (t 3 ). As the milling path  72  is incrementally interpolated using the process of  FIG.  20   , segments are traversed up until you reach the respective end of the milling path  72 , at which point the target pose TP no longer updates in that direction. If a firm end-of-path behavior is not desired (i.e., if it&#39;s desired for the user to “fall off” the ends), an alternate approach would be to turn off one or more of the virtual constraints when an end-of-path target is reached (as determined by the path handler  82 ), allowing the tool  20  to move freely. This behavior could be accompanied by user feedback on the displays  38 , audible feedback, and/or haptic feedback. 
     A user pendant, separate from the tool  20 , or other user interface UI (e.g., on the tool  20 ) may be employed to switch between the various modes of operation of the manipulator  14 . The control system  60  may be configured to automatically switch modes in certain situations. For example, if the control system  60  was operating the manipulator  14  in the semi-autonomous mode initially, prior to switching to the guided-manual mode, the control system  60  may automatically restart the semi-autonomous mode once the user switches off the guided-manual mode. The control system  60  may also first prompt the user before automatically continuing in the semi-autonomous mode, such as by providing selectable prompts on one or more of the displays  38  to continue in the semi-autonomous mode. The user may select to continue in the manual mode, semi-autonomous mode, etc. The control system  60 , if transitioning to the semi-autonomous mode from the guided-manual mode, may first calculate a transition path to the milling path  72 , which may be the shortest distance from the current position of the TCP to the milling path  72 , and then utilize this transition path as a form of lead-in path to resume movement along the milling path  72 . Upon this transition from the guided-manual mode to the semi-autonomous mode (or upon transition from the manual mode to the guided-manual mode), it may be desirable to re-compute the milling path  72  based on what volume of tissue was removed in the guided-manual mode and/or the manual mode (or alternatively, based on which volume of tissue is remaining), to have an optimized/cleaned up milling path  72 , reduced machining time, less air cutting, etc. Logging the volume of tissue removed, such as during the manual mode, in order to re-compute a milling path and improve efficiency in the semi-autonomous mode is described in U.S. Pat. No. 9,566,122, hereby incorporated herein by reference. Those same principles can be applied here when transitioning from the guided-manual mode to the semi-autonomous mode or from the manual mode to the guided-manual mode. 
     When the control system  60  is operating in the semi-autonomous mode and the user instructs the control system  60  to switch to the guided-manual mode, or when the control system  60  automatically switches to the guided-manual mode, the control system  60  records the last position on the milling path  72  that the tool  20  occupied in the semi-autonomous mode before switching to the guided-manual mode. After moving along the milling path  72 , in guided-manual mode as desired, the tool  20  is in a different position on the milling path  72  than the last recorded position. The control system  60 , when switching back to the semi-autonomous mode, can return back to the last recorded position to pick up where the semi-autonomous mode left off. In some versions, the lead-in path  72   a  described above may be used in the guided-manual mode to reach the starting point SP, which is the starting point SP for semi-autonomous machining. In this case, once the tool  20  is at the starting point SP, then semi-autonomous operation can start automatically or following user prompting and selection. 
     The current pose of the tool  20  relative to the milling path  72  and/or relative to the surgical site may be output by the navigation system  32  and represented on the displays  38  via graphical representations of the tool  20 , virtual boundaries  71 , milling path  72 , and/or the surgical site, e.g., the femur F or other anatomy. These graphical representations may update in real-time so that the user is able to visualize their progress in the guided-manual mode relative to the complete milling path  72  or portions of the milling path  72 . For example, the graphical representations of the tool  20  and anatomy may move on the displays  38  in real-time with actual movement of the tool  20  by the manipulator  14  and the anatomy. The milling path  72  can also be updated as described above to show the user the portions/segments of the milling path  72  already traversed by the tool  20  in the guided-manual mode. These can be differentiated from portions/segments that still need to be traversed, such as by rendering them different colors, line types, graying out or removing the traversed portions/segments, or the like. 
     The TCP of the tool  20 , and/or other portions of the tool  20 , may be additionally, or alternatively, constrained to the milling path  72  by employing virtual boundaries along and/or around the milling path  72 . Such virtual boundaries  90 ,  92  are shown in  FIGS.  24  and  25   . These virtual boundaries may represent tubes disposed about the milling path  72  with an entrance  94  and an exit  96 . The entrance  94  and/or the exit  96  of the virtual boundaries  90 ,  92  may be funnel-shaped or other shape to facilitate entrance of the tool  20  into the virtual boundary  90 ,  92 . These virtual boundaries  90 ,  92  may constrain movement of the tool in a manner similar to that disclosed in U.S. Pat. No. 9,119,655, incorporated herein by reference, or as described in U.S. Pat. No. 8,010,180, hereby incorporated herein by reference. 
     In this embodiment, the TCP of the tool  20  may be modeled as a sphere and a tube-shaped triangle mesh surface (with the tube having the same, approximately the same, or slightly larger diameter as the tool&#39;s sphere model) may be created that traverses the milling path  72 . This mesh surface can be created ahead of time (offline), but could be generated on-the-fly, such as by the boundary generator. Mesh density may be high enough (e.g., small enough triangles, typically 0.1 to 1.0 mm on a side) such that the triangle mesh sufficiently and smoothly represents a tube surface, resulting in a smooth feel as the tool  20  is slid down the tube. 
     When using these models in the guided-manual mode, velocity constraints are generated from each active mesh triangle (with each constraint acting along the normal vector of its corresponding mesh triangle) that prevent the tool  20  from escaping the tube once it enters. However, the tool  20  can slide freely in either direction down the length of the tube (constraints are normally not defined in these directions). The velocity constraints generated in this case would typically be passed into the constraint solver  86  as part of the boundary constraints output from the boundary handler  89 . Accordingly, in this embodiment, the path generator  68 , path handler  82 , and path constraint calculator  84  may be absent. Optionally, additional damping forces and/or damping constraints can be added to add/create damping along the tube surface if desired (e.g., for machining stability, feel, user preference, etc.). The constraints are passed into the constraint solver  86  and solved (possibly with other active constraints) by the constraint solver  86 , generating a constraint force Fc, which is summed by the virtual simulator along with other forces (F cgext , F damping , F inertial ) applied to the virtual rigid body, and forward dynamics performed to get a resulting commanded pose CP. 
     The tube can have a three-dimensionally curved path. This may be useful, for example, to assist in creating a helical motion for peg hole machining, e.g., allowing the user to use a small bur to more easily machine a larger diameter hole by following a helical path. A helical-shaped mesh can be generated having a much smaller diameter than the physical bur, to prevent the mesh from intersecting with itself, with the sphere representing the tool  20  having the same smaller diameter. By choosing an appropriate slope/pitch for the helical tube, the user only needs to apply gentle downward force and the tool  20  slides its way down the helix. The mesh can be created as shown in  FIG.  29    with a funnel shaped opening to guide the user to the entrance  94  of the tube and lead to the exit  96  that guides the user to pull the tool straight up following machining. 
     Referring to  FIG.  26   , the guided-manual mode described herein may be employed in various types of robotic systems. For example, the manipulator  14  may comprise a tele-manipulated robotic arm that is controlled via a user interface UI that is remotely located relative to the tele-manipulated robotic arm to control the tele-manipulated robotic arm. The user interface UI may comprise a separate manipulator such as a 6-DOF control unit that is manually manipulated by a user, e.g., a separate manipulator with active joints to provide haptic feedback to the user. During a free mode, the tele-manipulated robotic arm mimics the motion imparted to the user interface UI by the user. For example, the user interface UI may comprise the same linkage and joint configuration as the tele-manipulated robotic arm and movement of one or more links on the user interface UI cause movement of the corresponding links on the tele-manipulated robotic arm. In some embodiments, this coordinated movement may be at different scaling factors (e.g., a 1 mm movement of a link of the user interface UI may correlate to a 10 mm movement of the same link of the tele-manipulated robotic arm). 
     During the guided-manual mode, the tele-manipulated robotic arm is controlled by the manipulator controller  26  to keep the tool  20  on the milling path  72 . This may be accomplished by restricting available movements of the user interface UI in a manner that causes the tele-manipulated robotic arm to only be moved in a manner that keeps the tool  20  on the milling path  72 . Such restriction could be performed via the virtual constraints described above being applied to the user interface UI, and by extension such constraints would effectively be applied to the tele-manipulated robotic arm owing to the tele-manipulated robotic arm being cooperatively connected to the user interface UI to move in the same manner, albeit at potentially different scaling factors. Additionally, or alternatively, the user interface UI may be freely moved by the user, but the manipulator controller  26  effectively ignores those movements of the user interface UI that would otherwise correspond to moving the tool  20  off the milling path  72  and the manipulator controller  26  only reacts to corresponding movements from the user interface UI that would keep the tool  20  on the milling path  72 . 
     Several embodiments have been described in the foregoing description. However, the embodiments discussed herein are not intended to be exhaustive or limit the invention to any particular form. The terminology, which has been used, is intended to be in the nature of words of description rather than of limitation. Many modifications and variations are possible in light of the above teachings and the invention may be practiced otherwise than as specifically described.