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Support vector machine based optimal control for minimizing energy consumption of biped walking motions

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Abstract

A kind of support vector machine based optimal control (SVM-OC) approach for minimizing energy consumption of biped walking motions is proposed in this work. Different from existing learning controllers, a SVM controller is incorporated into an optimal controller for biped robots, which aims at minimize an energy-related cost function with three constraints of biped walking robots, including the system dynamics in the single support phase, the system dynamics in the impact phase, and the initial state of the biped. The controller is deduced under the condition of small sample sizes for the SVM. Main contributions of this paper include two aspects: First, a SVM-OC problem for minimizing energy consumption of biped walking motions is defined, which provides new clues to design a kind of optimal controller under the condition of unknown system dynamic model and small sample sizes. Secondly, derivation of the proposed SVM-OC problem is provided in detail, and simulation results demonstrate the advantage of the proposed method compared with conventional SVM control methods and neural network (NN) control methods.

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Abbreviations

θ :

joint angle

M:

inertia matrix

V:

matrix of centripetal acceleration and Coriolis terms

G:

gravity vector

τ :

input torque vector

Θ:

state vector

w :

weight vector

ζ i :

slack variable

N :

number of the training samples for SVM learning

i :

index of the samples

C :

penalty factor

φ(·):

nonlinear mapping function

b :

bias

λ 1(t):

vector of time-varying Lagrange multipliers

γ 1 :

vector of constant Lagrange multipliers

γ 2 :

vector of constant Lagrange multipliers

α i :

vector of constant Lagrange multipliers

K :

mixed kernel function

K poly :

polynomial kernel function

K poly :

RBF kernel function

q:

degree of the polynomial kernel

σ :

width of the RBF kernel.

a:

mixing coefficient of the mixed kernel

(y h, zh):

the position of the hip joint

(y a, za):

the position of the swinging ankle joint

p :

walking step length

d :

height of swinging ankle

2l e :

length of lower limbs

(xzmp, yzmp, 0):

coordinate of the zero moment point

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Authors and Affiliations

  1. Department of Electronic Engineering, Shunde Polytechnic, Foshan Guangdong, China, 528-300

    Liyang Wang

  2. Faculty of Automation, Guangdong University of Technology, Guangzhou Guangdong, China, 510-006

    Liyang Wang, Zhi Liu & Yun Zhang

  3. Faculty of Science and Technology, University of Macau, Macau, China, 999-078

    Chun Lung Philip Chen

  4. School of Information and Communication, Sungkyunkwan University, Suwon, South Korea, 440-746

    Sukhan Lee

Authors
  1. Liyang Wang
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  2. Zhi Liu
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  3. Chun Lung Philip Chen
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  4. Yun Zhang
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Correspondence to Liyang Wang.

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Wang, L., Liu, Z., Chen, C.L.P. et al. Support vector machine based optimal control for minimizing energy consumption of biped walking motions. Int. J. Precis. Eng. Manuf. 13, 1975–1981 (2012). https://doi.org/10.1007/s12541-012-0260-7

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  • DOI: https://doi.org/10.1007/s12541-012-0260-7

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