Patent Publication Number: US-2022219830-A1

Title: Systems and methods for a spring-augmented quadrotor for interactions with constrained environments

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     The present document is a non-provisional application that claims benefit to U.S. provisional patent application Ser. No. 63/091,114 filed on Oct. 13, 2020, which is herein incorporated by reference in its entirety. 
    
    
     FIELD 
     The present disclosure generally relates to quadcopter systems; and in particular, to a quadrotor with variable geometry and associated control model to accommodate improved navigation through constrained spaces. 
     BACKGROUND 
     Recent work on quadrotor systems demonstrated their applications for challenging tasks such as inspections of cluttered and occluded environments, aerial grasping and contact-based navigation, where a quadrotor interacts physically with the environment while navigating through it. Flying through such cluttered environments often requires the quadrotor to traverse through gaps smaller than its size, while simultaneously ensuring successful missions. 
     For a rigid quadrotor, flying through constrained spaces leads to research topics such as executing aggressive maneuvers or performing real-time trajectory re-planning to avoid cluttered areas using computer vision techniques. Alternatively, a quadrotor can have an adaptive morphology, where a folding mechanism helps the quadrotor fly through narrow apertures. In previous designs, additional actuators were needed to control the adaptive morphology, which increases the total weight of the system and decreases its power-to-weight ratio. In either case, different interactions bring significant challenges for vehicle stability when the quadrotor experiences unknown interactions with edges formed by passageways and gaps that the quadrotor may fly through. 
     It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a top view of a spring-augmented quadrotor system illustrating a passive adaptive morphology; 
         FIG. 2A  is an exploded view showing the quadrotor system of  FIG. 1 ; 
         FIG. 2B  is a close-up view showing a torsional spring mechanism of an arm of the quadrotor system of  FIG. 1 ; 
         FIG. 3A  is a diagram showing a coordinate system for dynamic modeling of the quadrotor system of  FIG. 1 ; 
         FIG. 3B  is a diagram showing passive folding with respect to the coordinate system of  FIG. 3A ; 
         FIG. 4A  is a diagram showing a moment of inertia model for the quadrotor system of  FIG. 1 ; 
         FIG. 4B  is a diagram showing a moment of inertia model for the quadrotor system of  FIG. 1 ; 
         FIGS. 5A-5F  are graphical representations showing trajectory tracking performance of a standard non-adaptive controller vs the adaptive controller of the quadrotor system of  FIG. 1 ; 
         FIG. 6  is a schematic showing adjustment of yaw for the quadrotor system of  FIG. 1 ; 
         FIG. 7  is a block diagram showing a controller of the quadrotor system of  FIG. 1 ; 
         FIGS. 8A-8D  are sequential photographs showing the quadrotor system of  FIG. 1  flying through a narrow passageway; 
         FIGS. 9A-9C  are sequential photographs showing the quadrotor system of  FIG. 1  flying through a gap between objects; 
         FIG. 9D  is a composite photograph showing the quadrotor system of  FIG. 1  flying through a passageway; 
         FIG. 10A  is a graphical representation showing a 3-D trajectory of the quadrotor system of  FIG. 1  traveling through a narrow gap; 
         FIG. 10B  is a graphical representation showing results of a flight through a 0.39 m×0.04 m gap compared with a desired yaw angle reference trajectory; 
         FIG. 11A  is a graphical representation showing a 3-D trajectory of the quadrotor system of  FIG. 1  traveling through a channel; and 
         FIG. 11B  is a graphical representation showing results of a flight through a channel compared with a desired yaw angle reference trajectory of the quadrotor system of  FIG. 1 . 
     
    
    
     Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims. 
     DETAILED DESCRIPTION 
     Various embodiments of a novel quadrotor with variable geometry suitable which allows the quadrotor to physically interact with cluttered environments and fly through narrow gaps and passageways are disclosed herein. The compliant and variable geometry quadrotor with passive morphing capabilities disclosed is designed may include torsional springs at every arm hinge to allow for rotation driven by external forces. In one embodiment, the quadrotor includes a dynamic model for control of the quadrotor as well as an adaptive controller for trajectory tracking as further described herein. A corresponding Lyapunov stability proof of attitude tracking is also presented. Further, an admittance controller is designed to account for changes in yaw due to physical interactions with the environment. Finally, the disclosed design for the quadrotor has been validated in flight tests with two setups: a small gap and a passageway. The experimental results have demonstrated the unique capability of the quadrotor in navigating through constrained narrow spaces. Referring to the drawings, an embodiment of a quadrotor is illustrated and generally indicated as  100  in  FIGS. 1-11 . 
     Referring to  FIG. 1  and  FIGS. 2A-2B , a quadrotor  100  as disclosed herein can interact with its environment and modify its configuration in response to external forces imposed by the environment. In general, the quadrotor includes a body frame  102  and plurality of arms  104  extending therefrom. In particular, the body frame  102  includes a base  122  and a top plate  121 , and a portion of each of the plurality of arms  104  extends from a position between the base  122  and top plate  121  as shown. 
     Each of the plurality of arms  104  includes a propeller ( 130 ) driven by at least one motor ( 160 ), and a propeller guard  132  protecting the propeller ( 130 ) from objects within the environment. A portion of each arm  104  is mounted over a connection member  124  defining a hinge or joint  112  extending between the base  122  and the top plate  121  accommodating in-plate rotary motion of the arm  104  along a plane defined by the hinge (rotary motion indicated and further described in  FIGS. 3A-3B and 4A-4B ). In other words, when in contact with an obstruction, the arms  104  rotate about the hinge defined by the connection member  124 . 
     In addition, a torsional spring ( 106 ) is attached along and/or to each hinge. The torsional spring ( 106 ) deflects to accommodate the in-plane rotary motion of the arm  104  away from an original position as the arm engages an obstruction and provides an angular return force to urge the arm  104  back to the original position.  FIG. 1  shows the quadrotor  100  in spring-loaded and unloaded configurations (unloaded configuration shown in phantom). When in contact with an obstruction, propeller guards  132  located at an end of each of the plurality of arms  104  of the quadrotor  100  contact the obstruction and act as bumpers, also forcing the associated arm  104  to rotate about its respective joint  112 ; this mechanical compliance results in passive morphing and is critical for vehicle stability of the quadrotor  100  during physical interactions with the environment. As such, the quadrotor  100  defines a compliant vehicle frame which adapts to tight spaces by physically interacting with the space. 
     Compared to prior work on morphing quadrotors, the quadrotor  100  employing the torsional springs  106  as described results in negligible weight (&lt;2% of the total weight) for achieving adaptive morphology. Hence, the power-to-weight ratio of the quadrotor  100  is not significantly affected. In addition, as indicated, the rotary motion of the arms  104  is passive and actuators are not required. 
     The rest of this disclosure discusses the details of the quadrotor  100  and its fabrication followed by a description of a model and system dynamics for varying geometry of the quadrotor  100 . In addition, the present disclosure discusses an adaptive controller  210  (low-level controller,  FIG. 7 ) for trajectory tracking and an admittance controller  260  ( FIG. 7 ) for flight through passageways as well as providing experimental results to validate the quadcopter  100  for various testing scenarios. 
     Quadrotor Design 
     In some embodiments, the body frame  102  and propeller guards  132  of the quadrotor  100  are 3D printed using a polymer such as polylactide or another suitable material, to ensure that the quadrotor  100  is lightweight and resilient to impact forces experienced during interactions with the environment. In some embodiments, the quadrotor  100  includes a base  122  and a top plate  121  operatively coupled to four arms  104 . Each arm  104  of the quadrotor  100  includes an arm member  140  associated with a respective torsional spring  106  and a propeller guard  131 . The torsional springs  106  are each attached to a respective joint  112  that connects a respective arm member  140  to the top plate  121 . As shown, each torsional spring  106  defines a first end  161  and a second end  162 , wherein the first end  161  being secured to the top plate  121  and the second end  162  being secured to the arm member  140  and/or a connection member  124 , such as a screw. This assembled engagement between these components adds a rotational degree of freedom (DoF) about the connection member  124  located within each joint ( FIG. 2A ) to each arm  104 , thereby giving each arm  104  the ability to passively fold when in-plane external forces are exerted on the quadrotor  100 . This passive variable geometry design enables the quadrotor  100  to fly through narrow gaps and passageways by physically interacting with the environment and allowing obstacles to rotate the arms  104  away from a standard position and still maintain control. Propeller guards  132  are attached to an outer portion  141  of each of the arm members  140  and are designed to ensure smooth transitions during such flights. The arms  104  are designed in such a way that adjacent motors  160  and propellers  130  associated with each arm  104  are staggered at different heights, thus preventing physical interference between the arms  104  when folding occurs. In some embodiments, the torsional spring  106  is lightweight (4 g), capable of sufficient deflection (120°), and compliant enough (0.21 N·m/rad) so that the arms  104  can be folded without requiring the quadrotor  100  to provide large thrust force when moving forward through narrow spaces. One embodiment of the quadrotor  100  shown has a total width of 41 cm and a folded width of 28 cm. The total width of the quadrotor may be in the range of about 30-100 cm and the folded width may be in the range of about 20-70 cm. 
     Model and System Dynamics 
     The notations used in this disclosure will be described in greater detail as well as the details of a simplified lumped mass model to calculate a position of center of gravity (CG) of the quadrotor  100  and a moment of inertia matrix of the quadrotor  100  at any instant in time. For an asymmetric and varying CG, the mapping of control inputs to individual motor thrusts changes, so a derivation of a control allocation matrix is further described herein. As shown in  FIG. 7 , the quadrotor  100  includes a computing system  200  for control of components of the quadrotor  100  in which the computing system  200  implements a low-level controller  210  and an admittance controller  260 . 
     Notation 
     For system dynamics and modeling, an inertial frame {i 1 , i 2 , i 3 } and a body frame {b 1 , b 2 , b 3 } are defined as shown in  FIG. 3A and 3B . The origin of this body fixed frame is located at the CG of the quadrotor  100 . A {q 1 , q 2 , q 3 } frame is also defined with its origin at the geometric center of the top view of the frame in order to calculate the relative position of the varying CG with respect to it. The q 1  axis is parallel to the line connecting motors 1 and 3 when the corresponding arms are in their equilibrium positions. Similarly, the q 2  axis is defined perpendicular to q 1  and lies in b 1 -b 2  plane, as shown in  FIGS. 4A and 4B . The q 3  axis is perpendicular to the q 1 -q 2  plane and is along the b 3  axis. Table I lists the notations, and (⋅)d is used to denote the desired quantity of (⋅) in this disclosure. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
            
               
                   
                 m ∈     
                 point mass representing each pair of motor, 
               
               
                   
                   
                 propeller, and corresponding arm model 
               
               
                   
                 M ∈     
                 mass of the center piece 
               
               
                   
                 m t  ∈     
                 total mass of the entire system 
               
               
                   
                 x ∈      3   
                 position of the vehicle in the inertial frame 
               
               
                   
                 υ ∈      3   
                 velocity of the vehicle in the inertial frame 
               
               
                   
                 J ∈      3×3   
                 moment of inertia matrix in the body fixed 
               
               
                   
                   
                 frame 
               
               
                   
                 Ω ∈      3   
                 angular velocity in the body frame 
               
               
                   
                 R ∈ SO(3) 
                 rotation matrix from body frame to the 
               
               
                   
                   
                 inertial frame 
               
               
                   
                 cg ∈      3   
                 position of center of gravity of vehicle in 
               
               
                   
                   
                 the {q 1 , q 2 , q 3 } frame 
               
               
                   
                 β 1 . β 2  ∈ S 1   
                 angles made by arms 2 and 4 respectively 
               
               
                   
                   
                 with the negative q 1  axis 
               
               
                   
                 f ∈     
                 control input of total thrust 
               
               
                   
                 τ ∈      3   
                 control moments for roll, pitch and yaw 
               
               
                   
                 γ ∈      4   
                 angles made by each arm with q 2  axis 
               
               
                   
                 r ∈      4   
                 distance from each in to the CG 
               
               
                   
                 h ∈     
                 height of motor pairs 2 and 4 in q 3   
               
               
                   
                 k τ  ∈     
                 spring constant 
               
               
                   
                   
               
            
           
         
       
     
     Expression for Moment of Inertia 
     Because the quadrotor  100  has a varying geometry while flying through passageways, it is necessary to compute the location of CG at any instant of time and the system&#39;s moment of inertia in order to compute correct control moments for trajectory tracking. The quadrotor  100  is modeled as an assemblage of a large sphere of mass M with diameter D and point masses of m units at a distance of I each from the center of the sphere. In addition, spheres 2 and 4 are at a height of h units below the motor pairs of 1 and 3. The inertia tensor of the whole body is calculated considering the {q 1 , q 2 , q 3 } frame with origin O as shown in  FIGS. 4A and 4B . 
     Without loss of generality, let arms 2 and 4 make an angle of β 1  and β 2  with the negative qi axis respectively as shown in  FIG. 4A and 4B . The coordinates for CG in {q 2 , q 1 , q 3 } denoted by (cg y , cg x , ch z ) as functions of β 1  and β 2  are given by: 
         cg =(−   l (sin β 1 −sin β 2 ),−   l (cos β 1 +cos β 2 ),2   h ),
 
     where 
     
       
         
           
             𝒞 
             = 
             
               
                 ( 
                 
                   m 
                   
                     M 
                     + 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
                 ) 
               
               . 
             
           
         
       
     
     Let J xx , J yy , J zz  denote the moment of inertia about the b 1 , b 2 , b 3  axis, respectively, then the time-varying moment of inertia of the system, J(β 1 (t), β 2 (t)), can be written as: 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               β 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           , 
                           
                             
                               β 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               J 
                               xx 
                             
                           
                           
                             
                               J 
                               xy 
                             
                           
                           
                             
                               J 
                               xz 
                             
                           
                         
                         
                           
                             
                               J 
                               yx 
                             
                           
                           
                             
                               J 
                               yy 
                             
                           
                           
                             
                               J 
                               yz 
                             
                           
                         
                         
                           
                             
                               J 
                               zx 
                             
                           
                           
                             
                               J 
                               zy 
                             
                           
                           
                             
                               J 
                               zz 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     and the formulation of each term is described in Table II. 
     
       
         
           
               
             
               
                 TABLE II 
               
               
                   
               
               
                 Moment of inertia calculation from angles β 1 , β 2  (in kg − m 2 ) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 J xx   
                 
                   
                     
                       
                         
                           
                             2 
                             
                               ? 
                             
                           
                           ⁢ 
                           
                             MR 
                             2 
                           
                         
                         + 
                         
                           M 
                           ⁡ 
                           
                             ( 
                             
                               
                                 cg 
                                 y 
                                 2 
                               
                               + 
                               
                                 cg 
                                 z 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   cg 
                                   y 
                                   2 
                                 
                                 + 
                                 
                                   cg 
                                   z 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           m 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                         l 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         β 
                                         1 
                                       
                                     
                                     - 
                                     
                                       cg 
                                       y 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     h 
                                     - 
                                     
                                       cg 
                                       z 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           m 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       l 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         β 
                                         2 
                                       
                                     
                                     - 
                                     
                                       cg 
                                       y 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     h 
                                     - 
                                     
                                       cg 
                                       z 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 J yy   
                 
                   
                     
                       
                         
                           
                             2 
                             
                               ? 
                             
                           
                           ⁢ 
                           
                             MR 
                             2 
                           
                         
                         + 
                         
                           M 
                           ⁡ 
                           
                             ( 
                             
                               
                                 cg 
                                 x 
                                 2 
                               
                               + 
                               
                                 cg 
                                 z 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           m 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     l 
                                     - 
                                     
                                       cg 
                                       x 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 cg 
                                 z 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           m 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       - 
                                       l 
                                     
                                     - 
                                     
                                       cg 
                                       x 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 cg 
                                 z 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                           l 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           β 
                                           1 
                                         
                                       
                                       - 
                                       
                                         cg 
                                         x 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       h 
                                       - 
                                       
                                         cg 
                                         z 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                           l 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           β 
                                           2 
                                         
                                       
                                       - 
                                       
                                         cg 
                                         x 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       h 
                                       - 
                                       
                                         cg 
                                         z 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 J zz   
                 
                   
                     
                       
                           
                         
                           
                             
                               
                                 
                                   
                                     2 
                                     5 
                                   
                                   ⁢ 
                                   
                                     MR 
                                     2 
                                   
                                 
                                 + 
                                 
                                   M 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         cg 
                                         x 
                                         2 
                                       
                                       + 
                                       
                                         cg 
                                         y 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   m 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         cg 
                                         y 
                                         2 
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             l 
                                             - 
                                             
                                               cg 
                                               x 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   m 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         cg 
                                         y 
                                         2 
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             
                                               - 
                                               l 
                                             
                                             - 
                                             
                                               cg 
                                               x 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   m 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 - 
                                                 l 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               sin 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 β 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               cg 
                                               y 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 - 
                                                 l 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               cos 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 β 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               cg 
                                               x 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 m 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             l 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             sin 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               β 
                                               2 
                                             
                                           
                                           - 
                                           
                                             cg 
                                             y 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                               l 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             cos 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               β 
                                               2 
                                             
                                           
                                           - 
                                           
                                             cg 
                                             x 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 J xy  = J yx   
                 Mcg y cg x  − mcg y (l − cg x ) − mcg y (−l − cg x ) + m(−l sin β 1  − cg y )(−l cos β 1  − cg x ) 
               
               
                 J yz  = J zy   
                 Mcg y cg z  + 2mcg y cg z  + m(−l sin β 1  − cg y )(h − cg z ) + m(l sin β 2  − cg y )(h − cg z ) 
               
               
                 J zx  = J xz   
                 Mcg x cg z  − m(l − cg x )cg z  − m(−l − cg x )cg z  + m(−l cos β 1  − cg x )(h − cg z ) + m(−l cos β 2  − cg x )(h − cg z ) 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           ? 
                         
                         ⁢ 
                         
                           indicates text missing or illegible when filed 
                         
                       
                     
                   
                 
               
            
           
         
       
     
     The moment of inertia calculated using (2) is used to design an adaptive controller  210  for trajectory tracking. This formulation is verified using a SolidWorks model as shown in Table III in Appendix A. The parameter values of the lumped mass model for the current design are: M=710 g, m=95 g, D=10 cm, I=12.5 cm, h=−3 cm. 
     Defining Control Inputs and Control Allocation 
     Assuming that the thrust produced by each propeller  130  of the quadrotor  100  is directly controlled and that the direction of thrust generated is normal to the quadrotor plane, the total thrust is the sum of thrusts produced by each propeller  130 , that is, f=  f i . Further, let τ 1 , τ 2  and τ 3  denote the pitch, roll and yaw moments respectively and γ i , i=1, . . . , 4 denote the angles made by each arm  104  with the positive b 2  axis as shown in  FIG. 4B . The moments can be calculated about b 1  and b 2  axes as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             τ 
                             1 
                           
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 f 
                                 1 
                               
                               ⁢ 
                               
                                 r 
                                 1 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 γ 
                                 3 
                               
                             
                             + 
                             
                               
                                 f 
                                 2 
                               
                               ⁢ 
                               
                                 r 
                                 2 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 γ 
                                 2 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 
                                   f 
                                   3 
                                 
                                 ⁢ 
                                 
                                   r 
                                   3 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
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     where the torque generated by each propeller  130  ( FIG. 1 ) contributes to the total yaw moment and is equal to (−1) i c τ f i  for a fixed constant c τ . The control allocation matrix (CAM) of the total thrust f and total moment τ can be determined as a mapping to individual motor thrusts, f 1 , f 2 , f 3  and f 4 , as 
     
       
         
           
             
               
                 
                   
                     
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     The determinant of this matrix is: 
       det(CAM)=2 c   τ ( r   1   r   2  sin(γ 1 −γ 2 )+ r   1   r   4  sin(γ 1 −γ 4 )+ r   2   r   3  sin(γ 3 −γ 2 )+ r   3   r   4  sin(γ 4 −γ 3 )).
 
     It is shown that when arms  104 B- 104 D are at the same location, this matrix is singular, implying that the quadcopter  100  will lose some degrees of freedom for full attitude control and need to find solutions for a reduced order system. For this disclosure, it is ensured that CAM does not become singular at any instant of time. 
     In the rest of this disclosure, it is assumed that control inputs to the system are total thrust f∈  and torque τ∈   3  and use (4) to calculate the individual thrust needed for each propeller  130 . 
     System Dynamics 
     The dynamics of the quadrotor  100  about the CG with control inputs f and τ are given by: 
         m   t   {dot over (v)}=m   t   ge   3   −fRe   3 , 
       {dot over (x)}=v, 
       {dot over (R)}=R{circumflex over (Ω)},
 
         J{dot over (Ω)}=τ−Ω×JΩ,   (5)
 
     where g=9.81 ms −2  denotes the acceleration due to gravity, e3 denotes the i 3  axis unit vector and the hat map {circumflex over ( )}·: R 3 -&gt;SO(3) is a symmetric matrix operator defined by the condition that {circumflex over (x)}y=x×y, ∀x, y∈   3 . 
     Controller Design 
     Referring to  FIG. 7 , the low-level attitude controller  210  for trajectory tracking and how the design is leveraged with an admittance controller  260  for flying through passageways will be discussed. 
     Adaptive Attitude Control 
     For trajectory tracking of a varying morphology, an adaptive attitude controller  210  is disclosed on the nonlinear configuration Lie group which accounts for the quadrotor&#39;s  100  varying moment of inertia. The desired b 1d , b 2d  and b 3d  axes are chosen in a similar fashion as that of a standard quadrotor. Now, errors in R and Ω are defined as: 
         e   Ω   =Ω−R   τ   R   d Ω d ,
 
         e   R =½( R   d   τ   R−R   τ   R   d ) v .  (6)
 
     The control moment τ∈   3  is selected as: 
       τ= J (− k   R   e   R   −k   Ω   e   Ω   −k   Ω     d     ė   Ω +ζ d )+Ω× JΩ   (7)
 
     for any positive constants k R , k Ω , k dΩ  and ζ d =−{circumflex over (Ω)}R τ {dot over (R)} d Ω d   − +R τ R d {dot over (Ω)} d . It is proven that with this controller  210 , the tracking error of attitude dynamics will converge to zero asymptotically. The estimated angular acceleration, {dot over (Ω)}, can be obtained by numerical differentiation of the estimated angular velocity. As numerical differentiation results in noisy output, the estimated angular acceleration is not utilized for implementation purposes. 
     Proof: The asymptotic stability for the attitude error is given in the appendix. 
     Simulations: The comparison results of simulations for a case where β 1 =30°, β 2 =30°, x d =[5 5 −4] τ  and b 1d =[1 0 0] τ  with the adaptive controller  210  and a standard controller which does not account for the varying J(β 1 (t), β 2 (t)) are shown in  FIGS. 5A-5F . It is noticed that for a short distance, the non-adaptive controller does not deviate significantly from the desired trajectory. However, if the morphing is persistent for a longer duration, with a non-adaptive control, there is significant deviation from the desired trajectory, whereas the adaptive controller  210  achieves accurate tracking. This is attributed to the shifted CG and the varying moment of inertia which is not accounted by a non-adaptive controller to generate the appropriate f and τ control signals. 
     Admittance Control 
     In this section, an admittance controller  260  in yaw to account for the physical forces acting on the quadrotor  100  in relatively smaller gaps and tunnels is disclosed. It is critical to replan the yaw setpoint because as the quadrotor  100  approaches the passageway, unforeseen interactions can lead to unintended yaw moments. In such scenarios, if the yaw set-point is not updated, the quadrotor  100  tries to correct the yaw repeatedly during its flight and is prone to multiple collisions which may lead to unsuccessful flights. To this end, the yaw admittance controller  260  is included, where an outer loop is added to the low level controller  210  to modify the yaw reference trajectory.  FIG. 6  demonstrates a scenario where a multi-copter flies through a constrained space. Upon entering, the vehicle experiences interactions and the current yaw changes. At this instance, the vehicle would try to reject the interaction which could result a jerky motion. Therefore, an admittance controller  260  is implemented to update the desired yaw according to the current yaw and comply with the interactions. The complete control structure is showed in  FIG. 7 . The control in Equation (8) is used to generate a new desired yaw: 
       M φ {umlaut over (φ)} d   +D   φ {dot over (φ)} d   +K   φ φ d =φ,  (8)
 
     where φ is the current yaw. The tuning parameters of M φ , D φ  and K φ  are chosen such that the dynamics of φ d  is critically damped to track changes in current yaw, to avoid delayed responses and oscillations in φ d  from over-damped and under-damped dynamics, respectively. 
     Experiments and Discussions 
     This section demonstrates the performance of the disclosed quadcopter  100  through experiments of i) flying through a gap and ii) flying through a passageway. 
     Hardware Setup 
     Experiments were conducted in an indoor drone studio at the Arizona State University. An Optitrack motion capture system with 17 high-speed cameras was utilized to obtain the position and heading of the vehicle. The 3-D position and current heading were transmitted to PIXHAWK, at 120 Hz for real-time feedback control. A high-level onboard computer  200  implementing was an Intel UP-board which ran the Robot Operating System (ROS) for communication with motion-capture system. A multi-threaded application was implemented for the admittance controller  260  and to enable the serial communication with the PIXHAWK. The quadrotor  100  was equipped with two additional IMUs, one underneath each arm, to obtain the arm bending angle. 
     The low-level attitude control algorithm  210  was implemented as described herein. A quaternion-based complementary filter was implemented for attitude estimation. A Kalman filter based algorithm was implemented for the low-level position estimation, and a cascaded P-PID control structure for the position control module. The admittance parameters (M φ , D φ , K φ ) were (0.01, 0.2, 1.0) for both the flight tests. 
     Flight Through a Gap 
     In a first flight test shown in  FIG. 9A , the quadrotor  100  autonomously flew through a gap. The gap was a pair of wooden beams  10 , 90 cm tall, attached to a fixed base plate as shown in  FIG. 9A . The two wooden beams  10  were 30 cm apart, which was less than the width of the quadrotor  100  (41 cm) as measured from the tip of one propeller guard  132  to the tip of the diametrically opposite propeller guard  132 . The base of the gap was located at (0, 0.5, 0) and the tip&#39;s location was (0, 0.5, 0.9). The quadrotor  100  started at location (0, 0, 0.8) and the target location was (0, 1.5, 0.8).  FIG. 10A  shows the trajectory of the quadrotor  100  during flight. The interaction between the quadrotor  100  and the gap lasts for about 0.8 seconds. It can be observed that, this interaction resulted in a loss of height by approximately 30 cm as well as a change in the heading (yaw) of the quadrotor  100 . The quadrotor  100  complied with the change in heading using the admittance controller  260  as described herein.  FIG. 10B  demonstrates the current yaw and desired yaw of the quadrotor  100  while in admittance mode. After the quadrotor  100  entered the gap, the current yaw of the quadrotor  100  changed due to interaction with the wooden beams  10 . The admittance controller  260  changed the yaw setpoint to comply with these interactions. After the quadrotor  100  exited the gap, it returned to the height setpoint while maintaining the desired yaw generated by the admittance control  260 . 
     Flight Through a Passageway 
     After successfully evaluating flight through a gap, the performance of the quadrotor  100  is evaluated using a passageway  20  where the vehicle continuously interacts with the environment. For the experimental setup, two acrylic sheets of 1 meter length are utilized to create the passageway with a width of 29 cm.  FIG. 9B  shows the video snapshots of the quadrotor  100  flying through the passageway  20 . The quadrotor  100  remains in the passageway for 1 second. The quadrotor  100  started at (0, 0, 0.8) and the targeted location was (0, 3.0, 0.8). The beginning of the passageway  20  is located at 80 cm from the origin along the y-axis. While successfully flying through the passageway  20 , the quadrotor  100  lost height by 15 cm. The loss in height can be attributed to potential near-wall aerodynamic effects and physical interactions with the walls of the passageway  20 . As shown in  FIG. 6  and explained herein, an admittance control  260  was implemented to change the desired yaw and comply with the interactions.  FIG. 11B  represents the yaw admittance of the quadrotor  100  under interactions. As the quadrotor  100  enters the narrow passageway  20 , the desired yaw changes according to the current yaw, which is the expected behavior of admittance control  260 .  FIGS. 8A-8D  sequentially depict the arms  104 A-D morphing and adapting to the narrow passageway  20 , while flying through it. 
     In this disclosure a novel quadrotor  100  with a passive folding mechanism which is capable of flying through gaps and passageways with dimensions smaller than its full body width while interacting with the environment. An adaptive controller  210  is included for trajectory tracking as well as a yaw admittance controller  260  in the outer loop to account for physical interactions during the flights. The mechanical complexity and added weight of the quadrotor  100  was low compared to existing morphing quadrotors. Finally, the quadrotor  100  was validated in flight tests through narrow apertures and tunnel-like environments. 
     It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto.