Abstract:
Certain embodiments of a system for reducing backlash include a member geared for rotation in first and second directions. In various implementations, a first motor causes rotation in the first direction with an output biased to preclude space between mating gear components in the first direction, and a second motor, which is mechanically independent of the first motor, causes rotation in the second direction with an output biased to preclude space between mating gear components in the second direction.

Description:
BACKGROUND 
     System components such as mechanical couplings, sensors, and actuators often require precision for applications in which the performance of feedback control systems is critical. For instance, precise control of an electro-mechanical system such as a robot is important for many applications in industrial automation, consumer products, and logistics and supply chain operations. A common mechanical coupling in robotics utilizes gear couplings or gearboxes. Gearboxes can be particularly useful because electromagnetic transducers can deliver very low torques at very high speeds. In applications, such as robotics, where bidirectional actuation with a gear reduction, a hydraulic servo valve, or a piezoelectric transducer, for instance, is often necessary, backlash, dead-zone, and hysteresis can severely limit the performance of feedback control systems. 
     Backlash in a gearbox is the amount of space between mating components of gears. By design, this space is built in to account for, inter alia, imprecision in gear manufacture and thermal expansion. In applications in which the gear couplings can be reversed, such as robotics, backlash of the mating components of the gears negatively affects the ability of the gears to translate input motion into instantaneous output motion. Moreover, this hard non-linearity prevents accurate positioning and may lead to chattering and limit-cycle type instabilities. This increases the wear and tear on gears, which, in turn, further increases backlash. 
     Dead-zone is a type of non-linearity in which the system does not respond with an output until the input reaches a certain level. Sometimes this type of non-linearity is intentional. For instance, systems may incorporate a small dead-zone to prevent constant operation and thus reduce wear and tear around a nominal operating point. Dead-zone is distinct from backlash and hysteresis in the sense that it does not have memory of previous operating conditions. 
     Backlash is a form of mechanical hysteresis due to space between mating components in a transmission, i.e., the “play” between parts. More generally, hysteresis indicates a state dependency on the history of the state of the system. An example of a system that demonstrates hysteresis is a spring that undergoes both elastic and plastic deformation. When compressed, due to the plastic deformation, the rest position of the spring will change depending on how far the spring was compressed. Thus, the state of the spring (rest position) depends on the history of the system. 
     In standard gear trains, backlash is inevitable due to the over-constrained nature of gears. Care can be taken to significantly reduce backlash by tight tolerancing, but at significant cost. Other solutions to backlash have been developed. In one approach, harmonic drive gearboxes have zero backlash, but suffer from high flexibility, resonance vibration, friction and structural damping nonlinearities, and are typically proprietary and expensive. In another approach, cycloidal gearboxes are another style of gearbox which remove backlash, and while they do not share the same durability issues as harmonic drives, they still tend to be proprietary and expensive. 
     In another approach, anti-backlash gears are used. In application, an anti-backlash gear mates with a standard gear. A typical arrangement consists of two concentric gears, with one gear rigidly held to the shaft and the other gear coupled through a spring with a set pre-tension. The gear teeth of both anti-backlash gears sandwich the standard gear, and depending on the direction of force, independently act as the load path of the gear train. Thus, as long as the pre-tension is not overcome, the anti-backlash gear effectively captures and tracks the previous gear in the gear train, and there is no backlash. When the pre-load is overcome, however, the position of the anti-backlash gear becomes a function of the loading, the spring constant, and the pre-load, which has been viewed as an undesirable behavior. 
     In still another approach, dual actuators are used. In a typical configuration, two or more drives are utilized in parallel connection to the final output stage. Backlash is reduced by small positional differences in the drive, effectively replicating the effect of anti-backlash gears. Active control and synchronization is generally required between the drives, however, to manage positional differences and prevent large undesirable torques. 
     Some applications can tolerate a compliant drive, and may—as in the case of series elastic actuators—even require it. A typical approach in such applications is to find a gearbox with an appropriate level of backlash and an independent compliant system, and combine them. Although the gearboxes are chosen based on their relationship between input motion and output motion, the whole system behaves with an input motion and an output force or torque. 
     Furthermore, techniques exist for computationally addressing system non-linearities such as backlash, hysteresis, and dead-zones. These approaches include implementing inverses of the non-linearities in the controllers or implementing adaptive, fuzzy, or neural network control techniques. However, in most cases the parameters of the non-linearities are unknown, and more generally, these techniques can be complicated and may not provide robust performance over a wide range of plant or environmental conditions. 
     What is needed, then, is a system for reducing or minimizing backlash, hysteresis, and dead-zones that is uncomplicated, cost-effective, and durable. 
     SUMMARY OF THE INVENTION 
     Generally, the invention relates to utilizing elastic elements in series with two parallel motor outputs to antagonistically bias their corresponding mechanical couplings so as to mitigate the negative effects of backlash, hysteresis, and dead-zone on closed loop control of the system. The motor outputs can be separated and biased in opposite directions by independently controlling the output torque of respective motors, such that gearboxes between the motor outputs and the motors are not in a backlash region. This utilization of two series elastic actuators in parallel for each joint can eliminate backlash by creating an opposing torque which pre-loads each gearbox, so that for either direction of output, at least one gearbox is in mesh. As the elastic elements are passive, they require torque feedback and an effective control system to generate torque offset. Some embodiments of the invention include modifying the feedback to the elastic actuators in response to changes in control references (e.g., computation) or environmental conditions (e.g., disturbances) by altering the transfer function of the torque controllers. The torque bias can be a function of system variables and the torque-offset profile can take many shapes. The invention thus includes switching between torque profiles in response to environmental context or control references to help mitigate backlash and maintain efficiency of the system. 
     In general, in one aspect, embodiments of the invention feature a system for reducing backlash in an apparatus with a single output member geared for rotation in first and second opposite directions. The system includes a first motor for causing rotation in the first direction with an output biased to preclude space between mating gear components in the first direction, and a second motor, which due to the compliant mechanical coupling between motors allows it to be controlled without state feedback from the first motor, for causing rotation in the second direction with an output biased to preclude space between mating gear components in the second direction. 
     In one embodiment, the output of the first motor is biased by a first intermediate mechanical component in series with the first motor output and includes a first torque-biasing elastic element, and the output of the second motor is biased by a second intermediate mechanical component in series with the second motor output and includes a second torque-biasing elastic element. The motor outputs are connected to a common output coupling, and the first and second intermediate mechanical components may include torque-biasing elastic elements that impart a directed pre-load torque to the associated motor output. The motors may be aligned along a common axis or may be coupled to their respective intermediate mechanical components in such a manner as to arbitrarily align each motor axis. In general, the intermediate components include or are coupled to respective gearboxes. The outputs of the intermediate mechanical components are connected via couplings in parallel to the common output that comprises the load. Furthermore, the intermediate components may also be connected to the motor output via couplings such as belts or cables. If desired, the motor shafts may be synchronized, resulting in the mechanical equivalent of a single motor. In general, any mechanical coupling between the motor and the output may introduce backlash, hysteresis, and/or dead-zone. These couplings may result in different forms of backlash, hysteresis, or dead-zone between each of the assemblies. 
     The system may include a sensor for detecting the position or displacement between the first and/or second intermediate mechanical component. The sensor may be, for example, a Hall Effect sensor used to implement a non-contact displacement sensor for measuring the deflection of the elastic element and thus can be used to determine the output torque of a single intermediate mechanical component. In some embodiments, each of the motors is configured for torque in a single direction and to avoid a zero-torque output where the output bias of each motor is operatively caused by the motor itself. The system may also include means for maintaining a common offset torque between the first and second motors. The motors may also use a position sensor attached to the motor shaft to provide a feedback signal for position based control that synchronizes the motion of the motors to maintain an initial torque bias through the elastic elements. Other displacement sensors include potentiometers, infrared proximity sensors, optical encoders, LVDS sensors, laser distance sensors (single-point version of LIDAR, radar, or sonar, etc.), and capacitive distance sensors. 
     The sensors may introduce system dead-zone and/or hysteresis into the feedback loop of the control system. For example, an analog slider potentiometer may utilize a mechanical wiper across a resistive element. When the wiper physically changes direction, backlash due to compliance in the physical system may manifest as hysteresis in the sensor measurement. 
     In general, in another aspect, embodiments of the invention feature a method for reducing backlash in an apparatus with a member geared for rotation in first and second opposite directions by first and second motors. The method includes independently biasing the output of the first motor to preclude space between mating gear components in the first direction and independently biasing the output of the second motor to preclude space between mating gear components in the second direction. The system is able to eliminate output backlash even when one of the intermediate components is within its backlash region as long as the output torque is maintained by the motor that is not within its backlash region. The invention thus provides methods for mitigating system imperfections or non-linearities, such as backlash, hysteresis, and dead-zone by applying an antagonistic force or torque biasing of intermediate system components (e.g., a sensor or an actuator) and/or heterogeneous system components. 
     In one embodiment, the method includes limiting total torque output to less than maximum output of either the first or the second motor. The method may also include maintaining a constant offset between torques delivered by each motor and allowing the gear components to enter and cross over their backlash region at a first threshold torque value. In some embodiments, the method includes holding one of the motors steady at a second threshold value while continuing to apply torque with the other motor and simultaneously applying torque with the first and second motors beyond the second threshold value. 
     These and other objects, along with advantages and features of the embodiments of the present invention herein disclosed, will become more apparent through reference to the following description, the accompanying drawings, and the claims. Furthermore, it is to be understood that the features of the various embodiments described herein are not mutually exclusive and can exist in various combinations and permutations, even if not made explicit herein. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       Other features and advantages of the present invention, as well as the invention itself, can be more fully understood from the following description of the various embodiments, when read together with the accompanying drawings, in which: 
         FIG. 1  schematically illustrates, in a block diagram, one actuator of a dual-actuator system for reducing backlash in accordance with one embodiment of the invention; 
         FIG. 2  schematically illustrates, in isometric view, a system for reducing backlash in accordance with one embodiment of the invention; 
         FIG. 3  schematically illustrates, in isometric view, an upper portion (i.e., one actuator) of the system illustrated in  FIG. 2 ; 
         FIG. 4  schematically illustrates, in isometric view, an enlarged area of the upper actuator illustrated in  FIG. 3 ; 
         FIG. 5  is a graph illustrating a torque-offset profile of a system for reducing backlash in accordance with one embodiment of the invention where no motors reach zero torque output (enter into their backlash region); 
         FIG. 6  is a graph illustrating a torque-offset profile of a system for reducing backlash in accordance with one embodiment of the invention where a constant torque offset is maintained between motors; 
         FIG. 7  is a graph illustrating a torque-offset profile of a system for reducing backlash in accordance with one embodiment of the invention where a motor crosses over at the second motor&#39;s saturation point; 
         FIG. 8  is a graph illustrating a torque-offset profile of a system for reducing backlash in accordance with one embodiment of the invention where a motor crosses over before the second motor&#39;s saturation point; and 
         FIG. 9  schematically illustrates, in a block diagram, a model of a system for reducing backlash in accordance with one embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     Generally, embodiments of the present invention reduce or minimize backlash by utilizing elastic elements to bias two separate intermediate mechanical outputs, which are driven by two separate mechanical inputs. Backlash may be defined as entry of a geared interface into an equilibrium band, which corresponds to the range of positions representing a zero torque output (i.e., the gear teeth are not in contact or out of mesh). Refer first to  FIG. 1 , which, for simplicity of illustration, conceptually depicts the components of a single actuator  100  with an input and an output and equipped, in accordance with an embodiment of the invention, to reduce backlash. The mechanical components corresponding to the blocks of  FIG. 1  are illustrated in  FIGS. 3 and 4 , and  FIG. 2  illustrates a complete dual-actuator system. 
     The actuator  100  includes a first motor  102   a , a first gearbox  104   a , a first compliant element (or intermediate mechanical component)  106   a , and an output  108  coupled in series. A gearbox position x g  and a compliant element position x c  are tracked, such that the difference between the two can be used to determine the position of the intermediate mechanical component or the deflection of an elastic element. The actuator  100  also includes an additional motor, gearbox, and compliant element coupled in the same configuration mirrored about the output  108 . 
     With reference to  FIGS. 2-4 , a system  200  for reducing backlash includes two motors  202   a ,  202   b , two gearboxes,  204   a ,  204   b , two intermediate mechanical components  206   a ,  206   b , and an output mechanical coupling  208 . The system  200  also includes a mechanical coupling  210  such as a joint (restricted to rotate around a line such that both the input and output mechanical couplings remain in a plane so as to implement a single degree of freedom), that provides a common axis for many of the components. Also pictured is an input mechanical coupling  212 , which the output mechanical coupling  208  moves about a rotational joint having single free rotational axis of freedom. The input mechanical coupling  212  may be coupled to the system  200  via a pair of mounting brackets  214   a ,  214   b.    
     The motors  202   a ,  202   b  are each independently operable in two directions and controllable by a control system. Using two motors allows for the implementation of several torque-offset profiles, as described in greater detail below in connection with  FIGS. 5-8 . The control system may control position or impedance with a cascaded control, whereby the chosen variable specifies the desired output torque based on the chosen torque-offset profile. The motors  202   a ,  202   b  may be disposed at opposite ends of the mechanical coupling  210 , thus aligning them along a common axis. Each motor  202   a ,  202   b  is coupled to the associated gearbox  204   a ,  204   b , respectively, and these may also be aligned along the mechanical coupling  210 . Each gearbox  204   a ,  204   b  is configured to transfer motion from the associated motor  202   a ,  202   b  to associated motor outputs  216   a ,  216   b . The gearboxes  204   a ,  204   b  and the motor outputs  216   a ,  216   b  are coupled via mating gear components (e.g., teeth on a gear). In the embodiment depicted in  FIGS. 2-4 , the associated motor outputs  216   a ,  216   b  are disks centered on the mechanical coupling  210 , and thus aligned with the motors  202   a ,  202   b  and the gearboxes  204   a ,  204   b . Each motor output  216   a ,  216   b  is coupled to the associated intermediate mechanical component  206   a ,  206   b ; these are described in detail below. 
     With reference to  FIG. 4 , it is to be understood that all components described with respect to the first intermediate mechanical component  206   a  have identical equivalents on the second intermediate mechanical component  206   b . The intermediate mechanical component  206   a  includes a belt  218   a , a pair of torque-biasing elastic elements  220   a , a sensor  222   a , and an output mount  224   a . The belt  218   a  frictionally engages the torque-biasing elastic elements  220   a , thereby coupling the motor output  216   a  to the elastic elements  220   a  via the belt mounts  226   a . The motor output  216   a  may be aligned along a common axis with the input of the elastic element  220   a . The belt  218   a  may be made of any material that is sufficiently flexible to conform to a surface of the motor output  216   a  and sufficiently rigid to avoid stretching when tensile forces are applied, such as steel or certain polymers. 
     The belt mounts  226   a  each have a protrusion  228  extending from a surface thereof and through complementary bores on the output mount  224   a . The bores are tightly dimensioned to allow for smooth sliding contact with the protrusions  228 . The protrusions  228  bias the springs such that when the protrusion  228   a  compresses its associated spring, the protrusion  228   b  will equally extend the other spring (since they are coupled through the band  218   a ), resulting in a net torque applied through the output mount  224   a —thereby thus biasing the output mount to move. 
     The elastic elements  220   a  may be linear springs with identical spring constants that are disposed around the protrusions  228  and held between a surface of the output mount  224   a  and sealing elements  230  (e.g., nuts) rigidly attached to an end of the protrusions  228 . A bottom surface of the output mount  224   a  is rigidly connected to the output  208 , such that any motion of the output mount  224   a  is transferred to the output  208  and vice-versa. 
     The sensor  222   a  detects the position of the intermediate mechanical component  206   a  (and thus the deflection of the elastic elements  220   a ). In one embodiment, the sensor  222   a  includes magnets  232  mounted on the motor output  216   a  and a Hall effect sensor  234   a  mounted on the output mount  224   a . In this arrangement, as the intermediate mechanical component  206   a  moves relative to the output mount  224   a , the magnetic field experienced by the Hall effect sensor  234   a  changes because of the change in proximity to the magnets  232 . A control system may receive the output of the Hall effect sensor  234   a  to convert a measurement into a position or deflection reading, that given the spring constants that relate deflection to force, can be used to determine the net output torque applied to  208 . 
     As previously mentioned, the system  200  may be operated to produce a variety of torque-offset profiles because two separate and individually controllable motors  202   a ,  202   b  are used. In one torque profile  300 , the first motor  202   a  and the second motor  202   b  are always applying torque in opposite directions (i.e., one always produces a positive and the other always produces a negative torque) and are configured such that neither motor  202   a ,  202   b  ever reaches zero output torque  302 , as illustrated in  FIG. 5 . Additionally, the total torque output of the system  200  may be limited to less than the maximum output  304  of either motor  202   a ,  202   b . When a desired command torque is in the direction of the first motor&#39;s  202   a  operation (e.g., a first direction), the first motor  202   a  is run at a higher torque than the second motor  202   b . In this situation, the first motor  202   a  is considered the active motor because it drives an intermediate mechanical output in the direction of the command torque, whereas the second motor  202   b  is considered the resistive motor because it drives an intermediate mechanical output in the direction opposite the command torque. As the motor  202   a  drives the first gearbox  204   a  and the first motor output  216   a  in the first direction, the belt  218   a  is rotated and equally compresses one of the elastic elements  220   a  and extends the other elastic element  220   a . As a result, the elastic elements  220   a  create a pre-load torque opposing the torque in the first direction, biasing the motor output  216   a  against the torque in the first direction to preclude space between mating gear components connecting the gearbox  204   a  and the motor output  216   a  in the first direction. Similarly, as the second motor  202   b  drives the gearbox  204   b  and the motor output  216   b  in a second direction, the belt  218   b  is rotated and equally compresses one of the elastic elements  220   b  and extends the other elastic element  220   b , so that the elastic elements  220   b  create a pre-load torque opposing the torque in the second direction—biasing the motor output  216   b  against the torque in the second direction to preclude space between mating gear components connecting the gearbox  204   b  and the motor output  216   b  in the second direction. Because the first motor  202   a  is run at a greater torque than the second motor  202   b , the output  208  will rotate in the first direction assuming no external loads at the output. The movement of the motor outputs  216   a ,  216   b  moves the magnets  232   a ,  232   b , varying the magnetic fields on the Hall effect sensors  234   a ,  234   b , which can provide values to a control system to calculate the position of the motor outputs  216   a ,  216   b  or the deflection of the elastic elements  220   a ,  220   b . The relationship between output torque and measured displacement is non-linear. Such non-linearities can be exploited (by mechanical choices in the sensor design and/or mounting, for example) in order to concentrate sensor resolution in the region of low output torque. This higher resolution may improve controllability or control performance near the backlash region. 
     To move the output  208  in the second direction, the second motor  202   b  is driven at a greater torque than the first motor  202   a . The process above is essentially repeated, though the second motor  202   b  is now the active motor and the first motor  202   a  is the resistive motor. When switching from the first direction to the second direction, there is no backlash because the mating gear components connecting the gear box  204   b  and the motor output  216   b  are already pre-loaded in the second direction by the elastic elements  220   b.    
     Another torque-offset profile  400 , where the motors  202   a ,  202   b  maintain a constant torque output offset, is illustrated in  FIG. 6 . The torque output offset may be maintained by a conventional control system controlling the operation of the motors  202   a ,  202   b . The motors  202   a ,  202   b  may cross over (i.e., change direction) to apply torque in a different direction. This can result in both motors  202   a ,  202   b  applying torque in the same direction, which increases the total torque that may be applied as compared to the previous torque-offset profile. This is ideal when the output torque is not likely to be required to quickly shift to the opposite direction. However, the crossover events may occur frequently depending on the command torque, potentially introducing backlash from within the motors  202   a ,  202   b  to the system  200 . The frequency of the crossover events may be reduced by increasing the amount of torque offset or through means that evaluate the external load to indicate when the external load is likely to cause crossovers and choosing to follow a profile such as  300  that does not have the potential for both motors to be in their backlash region at the same time. 
     Yet another torque-offset profile  500 , illustrated in  FIG. 7 , is a combination of the previous two profiles  300 ,  400 . The resistive motor follows the profile  300  (i.e., not crossing over) until the active motor reaches a first threshold value  502  (which may be a saturation point where the active motor is outputting a maximum torque), at which point the resistive motor follows the profile  400  (i.e., crossing over). 
     A further torque-offset profile  600 , illustrated in  FIG. 8 , is similar to the profile  500 . When the active motor reaches a second threshold value  602  (which may be before a saturation point), the active motor is held at the second threshold value  602  as the resistive motor crosses over and continues applying torque until it reaches the second threshold value  602 . Both motors are driven simultaneously beyond the second threshold value to apply torque up to a saturation point. 
       FIG. 9  schematically illustrates a hardware model  700  of the system embodiment  200 . The model  700  includes connections to a previous link (i.e., an input mechanical coupling  710 ) and to a subsequent link body (i.e., an output mechanical coupling  712 ). Each of the motors  715   a ,  715   b  acts independently to reduce the backlash at the gearboxes  725   a ,  725   b . The motors  715   a ,  715   b  are treated as physically attached to the previous link  710 , while the gearboxes  725   a ,  725   b  transfer the torque to the subsequent link  712  through the intermediate outputs  730   a ,  730   b  (which may be, for example, belts, cables, chains, a drive shaft, hydraulic couplings, etc.). 
     The control blocks  725 ,  727 ,  730   a , and  730   b  form a cascaded control system that provides the motors  715  with reference signals that are used to drive the motor&#39;s output. The torque controllers  730   a ,  730   b  use feedback from the deflection sensor and provide closed-loop control, whereby the outputs of the motors  715   a ,  715   b  are controlled to track the desired torque input to the torque controllers  730   a ,  730   b  respectively, from the backlash opposition controller  727 . In some embodiments, the torque controllers  730   a ,  730   b  utilize standard proportional integral derivative (PID) linear feedback control with a single input (desired total torque) and single output (motor drive reference signal) is employed. Other techniques suitable for implementing closed-loop control within the torque controllers  730   a ,  730   b  include optimal control, robust control, and adaptive control. 
     The backlash opposition controller  727  implements the torque offset profiles shown in  FIGS. 5 ,  6 ,  7 , and  8 . In other words, the backlash opposition controller  727  takes a total torque command as input from the joint impedance controller  725  and generates torque inputs to each of the torque controllers  730   a ,  730   b . The joint impedance controller  725  utilizes a control input such as desired joint angle or desired joint impedance (stiffness and damping) to generate a single control output (desired total torque). In some embodiments, the joint impedance controller  725  utilizes proportional derivative (PD) linear feedback control with a single input (desired joint angle) and single output (desired total torque). In this case, desired stiffness and damping are analogous to setting the proportional and derivative gains of the control loop, respectively. Other conventional techniques suitable for implementing closed loop control within  725  are also known in the art. 
     Typically, when coupling two stiff gear motors in parallel to provide direct opposition to eliminate backlash, binding or significant loss in overall gearbox efficiency occurs. Moreover, the efficiency loss may not be smooth over time and thus can manifest as “torque ripple” or binding. 
     One solution is to constrain the maximum torque of the motor so as to avoid exceeding the rated loads of the gearbox. To effectively implement suitable active constraints in control, high bandwidth motor torque or current control is usually required. 
     Large gear reductions result in limited back-drivability due to friction. Non-backdrivable gearboxes can still experience large torques due to environmental interactions. The use of elastic elements further reduces the negative effects of driving both mechanical outputs to the point of overload where the intermediate outputs become stationary, blocking the motor outputs from affecting the intermediate outputs, as if they were jammed. In this case, the motors no longer contribute to the transfer function between the mechanical load and the stages in the gearbox in series up to the point where the gear tooth mesh are within a static friction regime. 
     A mechanical solution is possible by introducing elastic elements in series with the motor, gear reduction, and load. This arrangement effectively filters impact forces and helps to reduce binding, dynamic frictional losses between gears, and gear tooth wear. 
     The series elastic implementation alters the impedance of the system and acts as a low pass filter for torque disturbances in addition to delivering protection to the gearbox from large forces (such as those during large accelerations of the external load). The intrinsic impedance of the coupling between the motors and the load is related to the passive impedance of the elastic elements. The control system exploits the intrinsic physical properties of the dual-motor SEA coupling. This forms a hybrid design, incorporating the dynamics of the physical system with a control aspect that provides several different operational modes and from the fact that many different positions of the motors can result in identical output torques. 
     In any of the profiles  300 ,  400 ,  500 ,  600 , by superimposing the torque outputs of the motors and sufficiently biasing them, the equilibrium bands of the subsystems (i.e., the motors) do not overlap. However, even in the dual-motor configuration, it is theoretically possible to experience the effects of backlash. The boundary of the backlash event is defined as when the torque delivered from either motor to the motor output approaches zero. This can happen when an external torque is large enough to accelerate the joint at a rate or to a velocity beyond the rate or velocity of the no-load speed at which the motor applying torque in the same direction can accelerate its associated gearbox. Inadequate sensing of torque can also cause a backlash event by reporting a non-zero torque when the gearbox is indeed applying zero torque. 
     To mitigate the risk of experiencing a backlash event in the dual motor configuration, a minimum and opposing torque level can be commanded to each of the motors. However, arbitrarily large opposing torque levels may reduce the efficiency of the system, reduce the dynamic range of force sensing, and increase friction and wear. Switching between torque profiles in response to environmental context or control performance can also help to mitigate this risk. 
     The terms and expressions employed herein are used as terms and expressions of description and not of limitation, and there is no intention, in the use of such terms and expressions, of excluding any equivalents of the features shown and described or portions thereof. In addition, having described certain embodiments of the invention, it will be apparent to those of ordinary skill in the art that other embodiments incorporating the concepts disclosed herein may be used without departing from the spirit and scope of the invention. Accordingly, the described embodiments are to be considered in all respects as only illustrative and not restrictive. Furthermore, the configurations described herein are intended as illustrative and in no way limiting. Similarly, although physical explanations have been provided for explanatory purposes, there is no intent to be bound by any particular theory or mechanism, or to limit the claims in accordance therewith.