Modifying CMAC adaptive control with weight smoothing in order to avoid overlearning and bursting

Abstract

This paper proposes a method to prevent overlearning (weight drift) and bursting in adaptive control using the cerebellar model arithmetic computer (CMAC). Traditional robust adaptive methods such as deadzone, projection, e-modification are not necessarily suitable for CMAC control. The proposed method relies on the idea of weight smoothing, where the difference between adjacent weights in the CMAC is penalized in the adaptation. Previously proposed weight smoothing schemes are only suitable for one or two-input CMACs and do not have stability guarantees. This work extends the method for use with n-input CMACs and develops the adaptive scheme within a Lyapunov framework to guarantee uniformly ultimately bounded signals. Simulations with a two-link, flexible-joint robot subject to sinusoidal disturbance demonstrate the performance and stability of the new method.

Appendices

Appendix 1: Robot model

The dynamics of the robot use a large-gear-ratio approximation [19]

$$\begin{aligned} \mathbf{M}(\theta _2) \ddot{\varvec{\theta }}= & {} -\mathbf{C}(\varvec{\theta },\dot{\varvec{\theta }}) \dot{\varvec{\theta }}- \mathbf{D}_{\theta } \dot{\varvec{\theta }} - \mathbf{K}_s(\varvec{\theta }- \varvec{\phi }) \nonumber \\& +\mathbf{g}(\varvec{\theta })+ \varvec{\tau }(t), \end{aligned}$$

(51)

$$\begin{aligned} \mathbf{J}\ddot{\varvec{\phi }}= & {} - \mathbf{D}_{\phi } \dot{\varvec{\phi }} - \mathbf{K}_s( \varvec{\phi }- \varvec{\theta }) + \mathbf{u}, \end{aligned}$$

(52)

where $\varvec{\theta }\in \mathfrak {R}^{2 x 1}$ contains the output angle positions, $\varvec{\phi }\in \mathfrak {R}^{2 x 1}$ contains the angles of the internal rotors, $\mathbf{M}\in \mathfrak {R}^{2 x 2}$ is the inertia matrix, $\mathbf{C}\in \mathfrak {R}^{2 x 2}$ contains Coriolis and centripetal terms, $\mathbf{g}\in \mathfrak {R}^{2x1}$ contains terms due to gravity, $\mathbf{J}= {\mathrm{diag}}(0.111, 0.2304)$  kg $\mathrm {m}^2$ contains the inertias of the rotors after large-gear reduction, $D_{\theta } ={\mathrm{diag}}(4.5,0.5)$, $D_{\phi } = {\mathrm{diag}}(0.0704,0.0282)$ Nms/rad contain friction terms, $\mathbf{K}_s={\mathrm{diag}}(5, 5)$  Nm s/rad contains the flexible-joint stiffness (assumed identical in this work), $\mathbf{u}\in \mathfrak {R}^{2 x 1}$ contains the motor torques (control signals), and $\varvec{\tau }(t)$ contains any external disturbances (which are assumed to be bounded). The inertia matrix is given by

$$\begin{aligned} \mathbf{M}_{11}= \,& {} m_1 c_1+m_2 (c_1^2 +c_2^2+2 l_1 c_2 \cos (\theta _2))+I_1 +I_2 ,\end{aligned}$$

(53)

$$\begin{aligned} \mathbf{M}_{21}= \,& {} \mathbf{M}_{22}= m_2 (c_2^2 + l_1+c_2 \cos (\theta _2) + I_2 ,\end{aligned}$$

(54)

$$\begin{aligned} \mathbf{M}_{22}= \,& {} m_2 c_2^2+ I_2, \end{aligned}$$

(55)

where $l_1=0.34293$ m is the length of link 1, $l_2=0.26353$ m is the length of link 2, $c_1= 0.1590$ m is the distance from the first joint to the first link’s center of mass, $c_2=0.0550$ m is the distance from the second joint to the second link’s center of mass, $m_1= 1.51$ kg is the mass of link 1, $m_2=0.87326$ kg is the mass of link 2, $I_1=0.0392$ kg $\mathrm {m}^2$ is the inertia of link 1 (including motor 2), and $I_2=0.0081$ kg $\mathrm {m}^2$ is the inertia of link 2. The Coriolis and centripetal terms are given by

$$\begin{aligned} \mathbf{C}= \begin{bmatrix} h \dot{\theta }_2&h (\dot{\theta }_2+\dot{\theta }_1) \\ - h \dot{\theta }_1&0 \end{bmatrix}, \end{aligned}$$

(56)

where

$$\begin{aligned} h= m_2 l_1 c_2 \sin (\theta _2). \end{aligned}$$

(57)

The planar motion of the robot lies on a $10^{\circ }$ slant and thus the gravitation terms are given by $\mathbf{g}=\mathbf{g}_{90^{\circ }} \sin (10^{\circ })$ where

$$\begin{aligned} \mathbf{g}_{90^0}=\begin{bmatrix} m_1 c_1 g_a C_{\theta _1}+ m_2[c_2 C_{\theta _1+\theta _2} +l_1 C_{\theta _1}] \\ m_2 g_a c_2 C_{\theta _1+\theta _2} \end{bmatrix} , \end{aligned}$$

(58)

where $C_{\theta }=\cos {\theta }$ and $g_a$ is the gravitational acceleration constant.

Appendix 2: Backstepping control

This control derivation assumes the spring joint stiffnesses are identical (for modifications to make the method robust to differences in spring constants see [10]). The control attempts to track the output angle. Defining the state variables $\mathbf{x}_1 = \varvec{\theta }, \mathbf{x}_2= \dot{\theta }, \mathbf{x}_3 = \varvec{\phi }, \mathbf{x}_4 = \dot{\varvec{\phi }}$ along with desired trajectory $\mathbf{x}_d = [\varvec{\theta }_d \; \dot{\varvec{\theta }}_d \; \ddot{\varvec{\theta }}_d]$ results in errors

$$\begin{aligned} \mathbf{e}_1 = \mathbf{x}_1-\varvec{\theta }_d, \quad \mathbf{e}_2 = \mathbf{x}_2 - \dot{\varvec{\theta }}_d. \end{aligned}$$

The auxiliary error defines the first backstepping error

$$\begin{aligned} \mathbf{z}_1 = \varvec{\varLambda }\mathbf{e}_1 + \mathbf{e}_2. \end{aligned}$$

(59)

where $\varvec{\varLambda }$ has positive gains on the diagonals. The system written in strict-feedback form becomes

$$\begin{aligned} \mathbf{K}_s^{-1} \mathbf{M}\dot{\mathbf{z}}_1&= \mathbf{K}_s^{-1} \varLambda \mathbf{e}_2 - \mathbf{K}_s^{-1} \mathbf{C}\dot{\varvec{\theta }} - \mathbf{K}_s^{-1} \mathbf{D}_{\theta }, \\&\quad - \,\mathbf{x}_1+ \mathbf{x}_3 - \mathbf{K}_s^{-1} \mathbf{M}\ddot{\varvec{\theta }}_d,\end{aligned}$$

(60)

$$\dot{\mathbf{x}}_3= \mathbf{x}_4,$$

(61)

$$\begin{aligned} \mathbf{J}\dot{\mathbf{x}}_4= & {} - \mathbf{D}_{\phi } - \mathbf{K}_s(\mathbf{x}_3 - \mathbf{x}_1) + \mathbf{u}. \end{aligned}$$

(62)

Following the backstepping procedure, the second and third steps of backstepping will use errors

$$\begin{aligned} \mathbf{z}_2&= \mathbf{x}_3 - \varvec{\alpha }_1, \end{aligned}$$

(63)

$$\begin{aligned} \mathbf{z}_3&= \mathbf{x}_4 - \varvec{\alpha }_2, \end{aligned}$$

(64)

and the virtual controls and control are chosen to be

$$\begin{aligned} \varvec{\alpha }_1 =\, &\mathbf{x}_1 - \varvec{\varGamma }_{1}(\varvec{\theta }_d,\dot{\varvec{\theta }}_d,\ddot{\varvec{\theta }}_d,\mathbf{e}_1,\mathbf{e}_2) \hat{\mathbf{w}}_{1} - \mathbf{G}_1 \mathbf{z}_1 , \end{aligned}$$

(65)

$$\begin{aligned} \varvec{\alpha }_2 =\, &\mathbf{x}_2 - \varvec{\varGamma }_{2}(\varvec{\theta }_d,\dot{\varvec{\theta }}_d,\ddot{\varvec{\theta }}_d,\varvec{\theta }_d^{(3)},\mathbf{e}_1,\mathbf{e}_2,\mathbf{x}_3) \hat{\mathbf{w}}_{2} -\mathbf{z}_1 \nonumber \\&- \mathbf{G}_2 \mathbf{z}_2, \end{aligned}$$

(66)

$$\begin{aligned} \mathbf{u}=&- \varvec{\varGamma }_{3} (\varvec{\theta }_d,\dot{\varvec{\theta }}_d,\ddot{\varvec{\theta }}_d,\varvec{\theta }_d^{(3)},\varvec{\theta }_d^{(4)}, \mathbf{e}_1,\mathbf{e}_2,\mathbf{x}_3,\mathbf{x}_4) \hat{\mathbf{w}}_{3}, \nonumber \\&-\mathbf{z}_2 - \mathbf{G}_3 \mathbf{z}_3. \end{aligned}$$

(67)

The weight updates using e-modification are

$$\begin{aligned} \dot{\hat{\mathbf{w}}}_{k}&= \beta (\varvec{\varGamma }_{{k}^\mathrm{T}} {\mathbf{z}}_{k} - \nu \Vert {\mathbf{z}}\Vert {\hat{\mathbf{w}}}_{k}) \quad \text {for }\, k=1,2,3 \end{aligned}$$

(68)

where $\beta$ is a positive constant adaptation gain. To show stability first note that since the joint stiffnesses are assumed to be identical then the matrix $\mathbf{K}_1^{-1} \mathbf{M}$ is symmetric. Note also that its eigenvalues are bounded from above and below. Thus a Lyapunov-like function for the backstepping procedure is

$$\begin{aligned} V =\, &\frac{1}{2} \mathbf{z}_1^\mathrm{T} \mathbf{K}_1^{-1} \mathbf{M}\mathbf{z}_1 + \frac{1}{2} \mathbf{z}_2^\mathrm{T} \mathbf{z}_2 + \frac{1}{2} \mathbf{z}_3^\mathrm{T} J \mathbf{z}_3 + \frac{1}{2 \beta } \sum _{k=1}^{3} \tilde{\mathbf{w}}_k^\mathrm{T} \tilde{\mathbf{w}}_k, \nonumber \\ \end{aligned}$$

(69)

The uniform ultimate boundedness of signals can be established as in Sect. 2.3.

The simulations use $\varvec{\varLambda }={\mathrm{diag}}(2,2)$ and control gains $\mathbf{G}_1=\mathbf{G}_2=\mathbf{G}_3={\mathrm{diag}}(2,2)$.

The CMACs are only provided with inputs $\varvec{\theta }_d,\dot{\varvec{\theta }}_d,\ddot{\varvec{\theta }}_d$ to avoid weight drift due to oscillating inputs. Thus, the error will be slightly larger than if a full design for a neural network was used. The cell sizes on each input are listed in Table 1.

Table 1 CMAC cell parameters