Bi-Articular Muscle Actuation Design for Robot Arms

Bi-Articular Muscle Actuation Design for Robot Arms V. Salvucci Y. Kimura S. Oh Y. Hori Hori-Fujimoto Lab, The University of Tokyo ICRA 2011 Workshop on Biologically-inspired Actuation, Shanghai

Outline 1 Bi-articularly Actuated Robot Arms 2 Actuator Redundancy Problem Traditional: Pseudo-inverse Matrix (2 norm) Our Solution: The norm Approach 3 Experimental Setup BiWi:Bi-Articularly Actuated & Wire Driven Robot Arm Feedforward Control Strategy 4 Experimental Results 5 Conclusions Bi-Articular Muscle Actuation Design for Robot Arms 2/24

What are Bi-articular Actuators? Multi-articular actuators produce torque in 2 (or more) consecutive joints Biceps brachii Simplified model of human musculo-skeletal structure Coracobrachialis Brachialis f 1 e 1: antagonistic pair of mono-articular muscles f 2 e 2: antagonistic pair of mono-articular muscles f 3 e 3: antagonistic pair of bi-articular muscles. Salvucci, Y. Kimura, S. Oh, Y. Hori Hori-Fujimoto Lab, The University of Tokyo i-articular Muscle Actuation Design for Robot Arms 4/24

Why Bi-Articular Actuators? 1 Homogeneous Maximum Force at End Effector [Fujikawa 1999] 2 Impedance control without FB [Hogan 1985] 3 Power transfer from proximal to distal joints [Schenau 1989] Bi-Articular Muscle Actuation Design for Robot Arms 5/24

Why Bi-Articular Actuators? 2 actuators of 10 Nm each 3 actuators of 6.6 Nm each Safety: smaller peak force (in case of controller failure) Vertical balance: greater ground horizontal force [Salvucci 2011b] 1 Homogeneous Maximum Force at End Effector [Fujikawa 1999] 2 Impedance control without FB [Hogan 1985] 3 Power transfer from proximal to distal joints [Schenau 1989] Bi-Articular Muscle Actuation Design for Robot Arms 6/24

Actuator Redundancy Problem Model Statics { T 1 = (f 1 e 1)r + (f 3 e 3)r T 2 = (f 2 e 2)r + (f 3 e 3)r { T 1 = τ 1 + τ 3 T 2 = τ 2 + τ 3 Given desired T 1 and T 2 τ 1=?, τ 2=?, τ 3=? Bi-Articular Muscle Actuation Design for Robot Arms 9/24

Pseudo-inverse Matrix (2 norm) Moore Penrose is the simplest pseudo inverse matrix = 2 norm [Klein 1983] 2 norm optimization criteria minimize subject to τ1 2 + τ 2 2 + τ 3 2 (1) { T 1 = τ 1 + τ 3 (2) T 2 = τ 2 + τ 3 Closed form solution τ 1 = 2 3 T1 1 3 T2 τ 2 = 1 3 T1 + 2 3 T2 (3) τ 3 = 1 3 T1 + 1 3 T2 T = [2.0, 1.5] τ = [1.66, 0.33, 0.83] Given F T = ( J T ) F T τ using (3). Salvucci, Y. Kimura, S. Oh, Y. Hori Hori-Fujimoto Lab, The University of Tokyo i-articular Muscle Actuation Design for Robot Arms 10/24

Our Solution: The norm Approach [Salvucci 2010] norm optimization criteria minimize max{ τ 1, τ 2, τ 3 } { (4) subject to T 1 = τ 1 + τ 3 T 2 = τ 2 + τ 3 (5) Closed form solution [Salvucci 2010] if T 1T 2 0 if T 1T 2 > 0 and T 1 T 2 if T 1T 2 > 0 and T 1 > T 2 τ 1 = T 1 T 2 2 τ 2 = T 2 T 1 (6) 2 τ 3 = T 1 +T 2 2 τ 1 = T 1 T 2 2 τ 2 = T 2 2 (7) τ 3 = T 2 2 τ 1 = T 1 2 τ 2 = T 2 T 1 2 (8) τ 3 = T 1 2 T = [2.0, 1.5] τ = [1.0, 0.5, 1.0] Given F T = ( J T ) F T τ using (6), (7), or (8) Bi-Articular Muscle Actuation Design for Robot Arms 11/24

Bi-articularly Actuated Robot Arms Actuator Redundancy Problem Experimental Setup Experimental Results Conclusions References BiWi: Bi-Articularly Actuated & Wire Driven Robot Arm [Salvucci 2011a] + Human-like actuation structure + Wire Transmission low link inertia (safety, energy efficiency) + Mono-/bi- articular torque decoupling (statics) - Not intrinsically compliant, but solvable with springs - Transmission loss in the wires V. Salvucci, Y. Kimura, S. Oh, Y. Hori Bi-Articular Muscle Actuation Design for Robot Arms Hori-Fujimoto Lab, The University of Tokyo 13/24

Feedforward Control Strategy F = [F x, F y ] T and T = [T 1, T 2 ] T : desired output forces and input torque. [τ 1,τ 2, τ 3 ]: desired actuator joint torques [e1, f1, e2, f2, e3, f3 ]: motor reference torques calculated as: { { ei Ktli τi if τi < 0 = f Ki τi if τi > 0 i = 0 otherwise 0 otherwise (9) where Ktl 2=1.33 (thrust wire transmission lost), Ktl 1 = K 3 = 0. F x and F y : measured forces at the end effector. Bi-Articular Muscle Actuation Design for Robot Arms 14/24

Bi-articularly Actuated Robot Arms Actuator Redundancy Problem Experimental Setup Experimental Results Conclusions References Infinity Norm VS Pseudo-inverse matrix (2 norm) [Salvucci 2011c] Measured maximum output force Relative difference in output force θ1 = 60 θ2 = 120 θ1 = 25 θ2 = 50 F diff = V. Salvucci, Y. Kimura, S. Oh, Y. Hori Bi-Articular Muscle Actuation Design for Robot Arms F n F 2 n F 2 n (10) Hori-Fujimoto Lab, The University of Tokyo 16/24

Conclusions Bi-articular muscles key points 1 Homogeneous distribution of output force 2 Power transfer proximal to distal joints 3 FF impedance control BiWi, Bi-articularly actuated and Wire driven Robot Arm Human-like actuation structure Low link-inertia Safety, efficiency Perfect decoupling between mono- and bi- articular actuator (statics) The norm approach for actuator redundancy resolution Closed form solution based on a piecewise linear function continuous in all the domain D = {T 1, T 2} Maximization of force at the end effector: +30% than 2 norm Applicable to systems with 3 inputs and 2 outputs Bi-Articular Muscle Actuation Design for Robot Arms 18/24

Thank you for your kind attention V. Salvucci Y. Kimura S. Oh Y. Hori www.hori.k.u-tokyo.ac.jp www.valeriosalvucci.com. Salvucci, Y. Kimura, S. Oh, Y. Hori Hori-Fujimoto Lab, The University of Tokyo i-articular Muscle Actuation Design for Robot Arms 21/24

2 norm Vs norm in 2D k, α and β are constant Equation with infinite solutions k = αx + βy x and y represent the motor torques bounded 2 norm norm minimize x 2 + y 2 minimize max { x, y } Comparison Solutions comparison Smaller solution space for 2 norm max{y, x } max{y 2, x 2} no solution for 2 norm!! Bi-Articular Muscle Actuation Design for Robot Arms 22/24

The Best Norm Output Force for θ 2 {30, 60, 90, 120, 150 } τ 1 + τ 2 + τ 3 for θ 2 = 90 norm 1 norm 2 norm min ( τ 1 + τ 2 + τ 3 ) min ( τ1 2 + τ 2 2 + τ 3 2 ) min max{ τ1, τ2, τ3 } τ 1 + τ 2 + τ 3 of norm > τ 1 + τ 2 + τ 3 of 2 norm τ 1 + τ 2 + τ 3 of 2 norm > τ 1 + τ 2 + τ 3 of 1 norm The best norm: switching between 1 norm, 2 norm and norm... but the system could not be stable due to discontinuity in torque patterns. Salvucci, Y. Kimura, S. Oh, Y. Hori Hori-Fujimoto Lab, The University of Tokyo i-articular Muscle Actuation Design for Robot Arms 23/24

References T. Fujikawa, T. Oshima, M. Kumamoto, and N. Yokoi. Output force at the endpoint in human upper extremities and coordinating activities of each antagonistic pairs of muscles. Transactions of the Japan Society of Mechanical Engineers. C, 65(632): 1557 1564, 1999. N. Hogan. The mechanics of multi-joint posture and movement control. Biological Cybernetics, 52(5):315 331, 1985. V. Salvucci, S. Oh, and Y. Hori. Infinity norm approach for precise force control of manipulators driven by bi-articular actuators. In IECON 2010-36th Annual Conference on IEEE Industrial Electronics Society, pages 1908 1913, 2010. V. Salvucci, Y. Kimura, S. Oh, and Y. Hori. BiWi: Bi-Articularly actuated and wire driven robot arm. In IEEE International Conference on Mechatronics (ICM), 2011a. V. Salvucci, Y. Kimura, S. Oh, and Y. Hori. Disturbance rejection improvement in Non-Redundant robot arms by bi-articular actuators. In Industrial Electronics (ISIE), IEEE International Symposium on, 2011b. V. Salvucci, Y. Kimura, S. Oh, and Y. Hori. Experimental verification of infinity norm approach for force maximization of manipulators driven by bi-articular actuators. In American Control Conference (ACC), 2011c. G. J. V. I. Schenau. From rotation to translation: Constraints on multi-joint movements and the unique action of bi-articular muscles. Human Movement Science, 8(4):301 337, Aug. 1989.. Salvucci, Y. Kimura, S. Oh, Y. Hori Hori-Fujimoto Lab, The University of Tokyo i-articular Muscle Actuation Design for Robot Arms 24/24