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TECHNICAL PAPERS

# Adaptive Control of Harmonic Drives

[+] Author and Article Information
Wen-Hong Zhu

Spacecraft Engineering, Space Technologies, Canadian Space Agency, 6767 route de l’Aeroport, Saint-Hubert, QC J3Y 8Y9, CanadaWen-Hong.Zhu@space.gc.ca

Erick Dupuis, Michel Doyon

Spacecraft Engineering, Space Technologies, Canadian Space Agency, 6767 route de l’Aeroport, Saint-Hubert, QC J3Y 8Y9, Canada

The dynamic properties of the link connected to the joint are fully reflected by the physical joint torque $τ$.

In Eq. 4, $[−g(τ)kc,g(τ)kc]$ represents a closed interval of real numbers ranging from $−g(τ)kc$ to $g(τ)kc$.

The choice of the cutoff frequency is based on the trade-off between reducing noise created by the numerical differentiation and extending the control bandwidth. Whereas a small cutoff frequency will decrease the control bandwidth, a large cutoff frequency will increase the noise level in the time derivatives of the measurement signals.

In view of 6,7,8 together with $u=τr$, $τod=0$, $IL=0$, $ts−=−∞$, and $ts+=+∞$, it yields $u=kp(q̇r−q̇)+kI∫(q̇r−q̇)dt=kp(q̇d−q̇)+(kpλ+kI)(qd−q)+kIλ∫(qd−q)dt$.

The desired position is a chirp signal of $0.2rad∕s$ varying from $0.01Hz$ to $10Hz$ in $100s$, passing through an integrator.

All the other joints have the similar profile. They are not presented here to save space.

J. Dyn. Sys., Meas., Control 129(2), 182-193 (Aug 11, 2006) (12 pages) doi:10.1115/1.2431813 History: Received October 20, 2005; Revised August 11, 2006

## Abstract

Aimed at achieving ultrahigh control performance for high-end applications of harmonic drives, an adaptive control algorithm using additional sensing, namely, the joint and motor positions and the joint torque, and their practically available time derivatives, is proposed. The proposed adaptive controller compensates the large friction associated with harmonic drives, while incorporating the dynamics of flexspline. The $L2∕L∞$ stability and the $L2$ gain-induced $H∞$ stability are guaranteed in both joint torque and joint position control modes. Conditions for achieving asymptotic stability are also given. The proposed joint controller can be efficiently incorporated into any robot motion control system based on either its torque control interface or the virtual decomposition control approach. Experimental results demonstrated in both the time and frequency domains confirm the superior control performance achieved not only in individual joint motion, but also in coordinated motion of an entire robot manipulator.

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## Figures

Figure 1

Joint assembly with harmonic drive

Figure 2

Test facility with two seven-joint robots

Figure 3

Position tracking control at 0.5Hz in four cases: A, the proposed control; B, adaptive friction compensation only; C, flexspline dynamics only; and D, PID control. The dashed lines represent the desired positions, and the solid lines represent the actual positions. Ten times of position tracking errors are also illustrated.

Figure 13

Joint 3 position control result of a coordinated robot motion involving all seven joints with a velocity of 0.001rad∕s. The top graph shows the position tracking control result in which the dashed line represents the desired joint trajectory and the solid line represents the actual joint position. The bottom graph shows the position tracking error.

Figure 12

Joint 3 position control result of a coordinated robot motion involving all seven joints with a maximum velocity of 0.4rad∕s. The top graph shows the position tracking control result in which the dashed line represents the desired joint trajectory and the solid line represents the actual joint position. The bottom graph shows the position tracking error.

Figure 11

Transfer functions of the joint position control q∕qd of case A, the proposed control, versus case D, PID control

Figure 7

Parameter adaptation corresponding to Fig. 6 in cases A, the proposed control, and B, adaptive friction compensation only. The top two graphs illustrate the estimates of the Coulomb friction with the solid lines representing k̂c(θ̂1) and the dashed lines representing ĉ(θ̂2). The bottom two graphs illustrate the estimates of the viscous friction coefficients with the solid lines representing k̂vp(θ̂3) and the dashed lines representing k̂vn(θ̂4).

Figure 6

Position tracking control at 5.0Hz in four cases: A, the proposed control; B, adaptive friction compensation only; C, flexspline dynamics only; and D, PID control. The dashed lines represent the desired positions, and the solid lines represent the actual positions.

Figure 5

Joint control torques corresponding to Fig. 3 in cases A, the proposed control; B, adaptive friction compensation only; and C, flexspline dynamics only.

Figure 4

Parameter adaptation corresponding to Fig. 3 in cases A, the proposed control, and B, adaptive friction compensation only. The top two graphs illustrate the estimates of the Coulomb friction with the solid lines representing k̂c(θ̂1) and the dashed lines representing ĉ(θ̂2). The bottom two graphs illustrate the estimates of the viscous friction coefficients with the solid lines representing k̂vp(θ̂3) and the dashed lines representing k̂vn(θ̂4).

Figure 10

Transfer functions of the joint position control q∕qd of case A, the proposed control, versus case C, flexspline dynamics only

Figure 9

Transfer functions of the joint position control q∕qd of case A, the proposed control, versus case B, adaptive friction compensation only

Figure 8

Joint control torques corresponding to Fig. 6 in cases A, the proposed control; B, adaptive friction compensation only, and C, flexspline dynamics only

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