An Adaptive Control Strategy for a Five-Fingered Prosthetic Hand

An Adaptive Control Strategy for a Five-Fingered Prosthetic Hand CHENG-HUNG CHEN Measurement and Control Engineering Research Center Dept. of Biological Sciences Idaho State University Pocatello, ID 8329 USA chenchen@isu.edu D. SUBBARAM NAIDU Measurement and Control Engineering Research Center Dept. of Electrical Engineering Idaho State University Pocatello, ID 8329 USA naiduds@isu.edu MARCO P. SCHOEN Measurement and Control Engineering Research Center Dept. of Mechanical Engineering Idaho State University Pocatello, ID 8329 USA schomarc@isu.edu Abstract: In this paper, an adaptive control strategy is developed for the 14 degrees of freedom (DOFs), fivefingered smart prosthetic hand with unknown mass and inertia of all the fingers. In particular, the forward and inverse kinematics of the system regarding the analytical relationship between the angular positions of joints and the positions and orientations of the end-effectors (fingertips) have been obtained using a desired orientation for three-link fingers. The simulations of the resulting adaptive controller with five-fingered prosthetic hand show enhanced performance. Key Words: adaptive control, prosthetic hand, hard control, five finger hand, feedback linearization, trajectory planning 1 Introduction Due to the extreme complexity of human hand, that has 27 bones, controlled by about 38 muscles to provide the hand with 22 degrees of freedom (DOFs), and incorporates about 17, tactile units of 4 different units, reproducing the human hand in all its various functions and appearance is still a challenging task [1]. Prosthetic hands have been built to replace human hands that can fully operate the various motions, such as holding, moving, grasping, lifting, twisting and so on [1 5]. However, about 35% of the users do not regularly use their prosthetic hands because of several reasons, including poor functionality of the presently available prosthetic hands and psychological problems. Thus, designing and developing an artificial hand which can mimics the human hand as closely as possible both in functionality and appearance can overcome these problems. Hard computing/control (HC) techniques can be used at lower levels for accuracy, precision, stability and robustness. HC comprises proportional-integralderivative (PID) control [6], optimal control [7, 8], adaptive control [9 11] etc. with specific applications to prosthetic devices. However, our previous work [11] for a two-fingered, thumb and index finger, prosthetic hand showed that adaptive controller can overcome overshooting and oscillation. However, a five-fingered prosthetic hand with adaptive control technique has not been developed yet. In this work, we first describe briefly the trajectory planning problem, human hand anatomy and the inverse kinematics for two-link thumb and the remaining three-link fingers (index, middle, ring and little). Next, the dynamics of the prosthetic hand is derived and feedback linearization technique is used to obtain linear tracking error dynamics. Then the adaptive controller is designed to minimize the tracking error. The simulation results show that the five-fingered prosthetic hand with the presented adaptive controller can grasp an object without overshooting and oscillation. Conclusions and future work are provided in the last section. 2 Modeling 2.1 Trajectory Planning and Inverse Kinematics The trajectory planning using cubic polynomial was discussed in our previous work [5, 6, 11, 12] for a two-fingered (thumb and index finger) smart prosthetic hand. Figure 1 shows that index finger, middle finer, ring finer and little finger contain three revolute joints in order to do the angular movements. Metacarpal-phalangeal (MCP) joint is located between metacarpal and proximal phalange bone; proximal and distal interphalangeal (PIP and DIP) joints separate the phalangeal bones. Thumb contains metacarpal-phalangeal (MCP) and interphalangeal (IP) joints. In this work, q j 1, qj 2 and qj 3 repre- ISSN: 1792-4235 45 ISBN: 978-96-474-214-1

LATEST TRENDS on SYSTEMS (Volume II) Figure 2: The Definition of Global Coordinate and Local Coordinates Figure 1: The Joints of Five-Finger Prosthetic Hand Reaching a Rectangular Rod a two-fingered (thumb and index finger) smart prosthetic hand. sent the angular positions (or joint angles) of the first joint M CP j, the second joint P IP j and the third joint DIP j of index finger (j = i), middle finger (j = m), ring finger (j = r) and little finger (j = l), respectively; q1t and q2t are the angular positions of the first joint M CP t and the second joint IP t of thumb (t). Forward and inverse kinematics of articulated systems study the analytical relationship between the angular positions of joints and the positions and orientations of the end-effectors (fingertips). A desired trajectory is usually specified in Cartesian space and the trajectory controller is easily performed in the joint space. Hence, to convert Cartesian trajectory planning to the joint space [13] is necessary. Using inverse kinematics, the joint angular positions of each finger need to be obtained from the known fingertip positions (joint space). Then the angular velocities and angular accelerations of joints can be obtained from the linear and angular velocities and accelerations of fingertips (end-effectors) by the geometric Jocobian. As shown in Figure 2, X G, Y G, and Z G are the three axes of the global coordinate. The local coordinate xt -y t -z t of the thumb can be reached by rotating through angles α and β to X G and Y G of the global coordinate, subsequently. The local coordinate xi -y i -z i of index finger can be obtained by rotating through angle α to X G and then translating the vector di of the global coordinate; similarly, the local coordinate xj -y j -z j of middle finger (j = m), ring finger (j = r), and little finger (j = l) can be obtained by rotating through angle α to X G and then translating the vector dj (j = m, r and l) of the global coordinate. The inverse kinematics of two-link thumb and three-link fingers was discussed in our previous publications [5,6,11,12] for ISSN: 1792-4235 2.2 Dynamics of Hand The dynamic equations of hand motion are derived via Lagrangian approach using kinetic energy and potential energy as [5] d dt L q L = τ, q (1) where L is the Lagrangian; q and q represent the angular velocity and angle vectors of joints, respectively; τ is the given torque vector at joints. The Lagrangian L can be expressed as L = T V, (2) where T and V denote kinetic and potential energies, respectively. Substitute (2) into (1) and dynamic equations of thumb can be obtained as below. M(q)q + C(q, q ) + G(q) = τ, (3) where M(q) is the inertia matrix; C(q, q ) is the Coriolis/centripetal vector and G(q) is the gravity vector. (3) can be also written as M(q)q + N(q, q ) = τ, (4) where N(q, q )=C(q, q )+G(q) represents nonlinear terms. The dynamic relations for the two-link thumb and the remaining three-link fingers are quite lengthy and omitted here due to lack of space [5]. 46 ISBN: 978-96-474-214-1

3 Control Techniques 3.1 Feedback Linearization The nonlinear dynamics represented by (4) is to be converted into a linear state-variable system by finding a transformation using feedback linearization technique [1]. Alternative state-space equations of the dynamics can be obtained by defining the position/velocity state x(t) of the joints as x(t) = [ q(t) q(t) ]. (5) Let us repeat the dynamical model and rewrite (4) as d dt q(t) = M(q(t)) 1 [N(q(t), q(t)) τ(t)]. (6) Thus, from (5) and (6), we can derive a linear statevariable equation in Brunovsky canonical form as ẋ(t) = [ I ] x(t) + [ I with its control input vector given by ] u(t) (7) u(t) = M(q(t)) 1 [N(q(t), q(t)) τ(t)]. (8) Let us suppose the prosthetic hand is required to track the desired trajectory q d (t) described under path generation or tracking. Then, the tracking error e(t) is defined as e(t) = q d (t) q(t). (9) Here, q d (t) is the desired angle vector of joints and can be obtained by trajectory planning [5, 6, 11, 12]; q(t) is the actual angle vector of joints. Differentiating (9) twice, to get ė(t) = q d (t) q(t), ë(t) = q d (t) q(t). (1) Substituting (6) into (1) yields ë(t) = q d (t) + M(q(t)) 1 [N(q(t), q(t)) τ(t)] (11) from which the control function can be defined as u(t) = q d (t) + M(q(t)) 1 [N(q(t), q(t)) τ(t)].(12) This is often called the feedback linearization control law, which can also be inverted to express it as τ(t) = M(q(t)) [ q d (t) u(t)) + N(q(t), q(t)].(13) Using the relations (1) and (12), and defining state vector x(t) = [e(t) ė(t) ], the tracking error dynamics can be written as [ ] [ ] I ẋ(t) = x(t) + u(t). (14) I Note that this is in the form of a linear system such as ẋ(t) = Ax(t) + Bu(t). (15) 3.2 Adaptive Control Technique The tracking error e and the filtered tracking error r are defined as e = q d q, (16) r = ė + Λe. (17) Here, q d is the desired angle vector of joints; q is the actual angle vector of joints; Λ is the positive-definite diagonal gain matrix. The filtered error (17) ensures stability of the overall system so that the tracking error (16) is bounded. Figure 3 shows the block diagram of the adaptive controller. Here, the filtered signal r(t) is derived from the tracking error e(t) and the trajectory planner and is fed to the adaptive controller of the prosthetic hand. r( t) Closed-Loop Adaptive Controller Uncertainity Disturbance + +. N( q( t), q( t) ) Linear System f( t) = Y + + ( t) Smart q( t), q. K ( t) D Hand + + Closed-Loop PD Controller Trajectory Planner. e( t) +. q d ( t) - q d ( t) + - Figure 3: Block Diagram of Adaptive Control Technique Differentiating and substituting (17) into (3) gives the dynamic equation in terms of the filtered error r as. q( t) M(q)ṙ = C m (q, q)r + f τ (18) where C(q, q) = C m (q, q) q and the nonlinear term f can be defined as f = M(q)( q d + Λė) + C m (q, q)( q d + Λe) +G(q) + τ dis = Yπ. (19) Here, the regression matrix Y is a matrix of known robot functions and π is a vector of unknown parameters [9]. The regression matrix Y and the unknown parameter vector π of two-link thumb and three-link index finger are expressed in [5]. The torque vector τ can be calculated by e( t) q( t) τ = Yπ + K D r. (2) ISSN: 1792-4235 47 ISBN: 978-96-474-214-1

The unknown parameter rate vector π can be updated by π = Γ 1 Y r (21) where Γ is a tuning parameter diagonal matrix. 4 Simulation Results and Discussion When thumb and the other four fingers are doing extension/flexion movements, the workspace of fingertips is restricted to the maximum angles of joints. Referring to inverse kinematics, the first and second joint angles of the thumb fingertip are constrained in the ranges of [,9] and [-8,] (degrees). The first, second, and third joint angles of the other four fingers are constrained in the ranges of [,9], [,11] and [,8] (degrees), respectively [14]. Next, we present simulations with an adaptive controller for the 14 DOFs fivefingered smart prosthetic hand. The parameters of the two-link thumb/three-link index finger [15] were related to desired trajectory. All parameters of the smart prosthetic hand selected for the simulations are given in Table 1 and the side length and length of the target rectangular rod are.1 and.1 (m), respectively. The relating parameters between the global coordinate and the local coordinates are defined in Table 2. Besides, all links are assumed as a circular cylinder with the radius (R).1 (m), so the inertia I j zzk of each link k of each finger j (j = t, i, m, r, and l) can be calculated as I j zzk = 1 4 mj k R2 + 1 3 mj k Lj k2. (22) All initial actual angles are zero. Figure 4 to Figure 8 are the tracking errors and desired/actual angles of thumb, index finger, middle finger, ring finger, and little finger for the proposed five-fingered smart prosthetic hand. 5 Conclusions and Future Work An adaptive control strategy was developed for the 14 degrees of freedom (DOFs), five-fingered smart prosthetic hand with unknown mass and inertia of all the fingers. Further, the forward and inverse kinematics of the system regarding the analytical relationship between the angular positions of joints and the positions and orientations of the end-effectors (fingertips), was obtained using a desired orientation for three-link fingers. The simulations of for the resulting adaptive controller with five-fingered prosthetic hand showed good agreement between the reference and the actual trajectories. Work is in progress for developing an Table 1: Parameter Selection of the Smart Hand Parameters Values Thumb Time (t,t f ), 2 (sec) Desired Initial Position.35,.6 (m) Desired Final Position.495,.6 (m) Desired Initial Velocity, (m/s) Desired Final Velocity, (m/s).4,.4 (m) Index Finger Desired Initial Position.65,.8 (m) Desired Final Position.1,.6 (m).4,.4,.3 (m) Middle Finger Desired Initial Position.65,.8 (m) Desired Final Position.5,.6 (m).4,.4,.3 (m) Ring Finger Desired Initial Position.65,.8 (m) Desired Final Position.1,.6 (m).4,.4,.3 (m) Little Finger Desired Initial Position.55,.8 (m) Desired Final Position.2,.6 (m).4,.4,.3 (m) All fingers use same parameters Local coordinates adaptive/robust controller for the five fingered hand with 14-DOFs. Acknowledgements: The financial support for this research from the Telemedicine Advanced Technology Research Center (TATRC) of the U.S. Department of Defense (DoD) is gratefully acknowledged. References: [1] M. Zecca, S. Micera, M. Carrozza, and P. Dario, Control of multifunctional prosthetic hands by processing the electromyographic signal, Critical Reviews TM in Biomedical Engineering, vol. 3, pp. 459 485, 22, (Review article with 96 references). [2] J. C. K. Lai, M. P. Schoen, A. Perez-Gracia, D. S. Naidu, and S. W. Leung, Prosthetic devices: Challenges and implications of robotic implants and biological interfaces, Proceedings of the Institute of Mechanical Engineers ISSN: 1792-4235 48 ISBN: 978-96-474-214-1

Table 2: Parameter Selection of the Relation between Global and Local Coordinates Parameters α β d i d m d r d l 1 Values 9 (degrees) 45 (degrees) (.35,, ) (m) (.4,, -.2) (m) (.35,, -.4) (m) (.25,, -.6) (m) 1 3 25 2 15 1 5 Joint 1 of Index Finger Joint 2 of Index Finger Joint 3 of Index Finger 5 5 1 15 2 1 9 8 7 6 5 4 3 2 1 5 1 15 2 8 6 4 2 2 4 Joint 1 of Thumb Joint 2 of Thumb 6 5 1 15 2 8 6 4 2 2 4 6 5 1 15 2 Figure 4: Tracking Errors (left) and Joint Angles (right) for Adaptive Controller of Thumb (IMechE), Part H: Journal of Engineering in Medicine, vol. 221, no. 2, pp. 173 183, January 27, special Issue on Micro and Nano Technologies in Medicine. [3] L. Zollo, S. Roccella, E. Guglielmelli, M. C. Carrozza, and P. Dario, Biomechatronic design and control of an anthropomorphic artificial hand for prosthetic and robotic applications, IEEE/ASME Transactions on Mechatronics, vol. 12, no. 4, pp. 418 429, August 27. [4] D. S. Naidu, C.-H. Chen, A. Perez, and M. P. Schoen, Control strategies for smart prosthetic hand technology: An overview, in Proceedings of the 3th Annual International IEEE EMBS Conference, Vancouver, Canada, August 2-24 28, pp. 4314 4317. [5] C.-H. Chen, Hybrid control strategies for smart prosthetic hand, Ph.D. dissertation, Measurement and Control Engineering, Idaho State University, May 29. [6] C.-H. Chen, D. S. Naidu, A. Perez, and M. P. Schoen, Fusion of hard and soft control techniques for prosthetic hand, in Proceedings of Figure 5: Tracking Errors (left) and Joint Angles (right) for Adaptive Controller of Index Finger 6 5 4 3 2 1 Joint 1 of Middle Finger Joint 2 of Middle Finger Joint 3 of Middle Finger 1 5 1 15 2 12 1 8 6 4 2 2 5 1 15 2 Figure 6: Tracking Errors (left) and Joint Angles (right) for Adaptive Controller of Middle Finger the International Association of Science and Technology for Development (IASTED) International Conference on Intelligent Systems and Control (ISC 28), Orlando, Florida, USA, November 16-18 28, pp. 12 125. [7] D. Naidu, Optimal Control Systems. Boca Raton, FL: CRC Press, 23. [8] C.-H. Chen, D. S. Naidu, A. Perez-Gracia, and M. P. Schoen, A hybrid control strategy for five-fingered smart prosthetic hand, in Joint 48th IEEE Conference on Decision and Control (CDC) and 28th Chinese Control Conference (CCC), Shanghai, P. R. China, December 16-18 29, pp. 512 517. [9] F. Lewis, S. Jagannathan, and A. Yesildirek, Neural Network Control of Robotic Manipulators and Nonlinear Systems. London, UK: Taylor & Francis, 1999. ISSN: 1792-4235 49 ISBN: 978-96-474-214-1

3 25 2 15 1 5 Joint 1 of Ring Finger Joint 2 of Ring Finger Joint 3 of Ring Finger 5 5 1 15 2 1 9 8 7 6 5 4 3 2 1 5 1 15 2 [13] B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo, Robotics: Modelling, Planning and Control. London, UK: Springer-Verlag, 29. [14] P. K. Lavangie and C. C. Norkin, Joint Structure and Function: A Comprehensive Analysis, Third Edition. Philadelphia, PA: F. A. Davis Company, 21. [15] S. Arimoto, Control Theory of Multi-fingered Hands: A Modeling and Analytical-Mechanics Approach for Dexterity and Intelligence. London, UK: Springer-Verlag, 28. Figure 7: Tracking Errors (left) and Joint Angles (right) for Adaptive Controller of Ring Finger 45 4 35 3 25 2 15 1 5 Joint 1 of Little Finger Joint 2 of Little Finger Joint 3 of Little Finger 9 8 7 6 5 4 3 2 1 5 5 1 15 2 5 1 15 2 Figure 8: Tracking Errors (left) and Joint Angles (right) for Adaptive Controller of Little Finger [1] F. Lewis, D. Dawson, and C. Abdallah, Robot Manipulators Control: Second Edition, Revised and Expanded. New York, NY: Marcel Dekker, Inc.,, 24. [11] C.-H. Chen, D. S. Naidu, A. Perez-Gracia, and M. P. Schoen, A hybrid adaptive control strategy for a smart prosthetic hand, in The 31st Annual International Conference of the IEEE Engineering Medicine and Biology Society (EMBS), Minneapolis, Minnesota, USA, September 2-6 29, pp. 556 559. [12] C.-H. Chen, D. Naidu, A. Perez-Gracia, and M. P. Schoen, A hybrid optimal control strategy for a smart prosthetic hand, in Proceedings of the ASME 29 Dynamic Systems and Control Conference (DSCC), Hollywood, California, USA, October 12-14 29, (No. DSCC29-257). ISSN: 1792-4235 41 ISBN: 978-96-474-214-1