Control of Robot Manipulators in Joint Space R. Kelly, V. Santibanez and A. Loria

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.d12.2 PD Control plus Feedforward 275�where tanh(q) is the vectorial tangent hyperbolic function (12.10) and � >0is a given constant.To show that the Lyapunov function candidate (12.14) is positive definiteand radially unbounded, we first observe that the third term in (12.14) satisfiesM(q)q� �� tanh(q)T M(q)q d" � tanh(q) �� d" � �Max{M} tanh(q) q�� d" � �1�Max{M} q q� �where we used tanh(q) d"�1 q in the last step.From this we obtainq� �� -�tanh(q)T M(q)q e"-� �1�Max{M} q �.Therefore, the Lyapunov function candidate (12.14) satisfies the followinginequality:�# �#T �# �#� �q q�min{Kp} -� �1 �Max{M}1��# �# �# �#� �V (t, q, q) e"q q2 � ��-� �1 �Max{M} �min{M} �and consequently, it happens to be positive definite and radially unboundedsince by assumption, Kp is positive definite (i.e.�min{Kp} > 0) and we alsosupposed that it is chosen so as to satisfy (12.13).Following similar steps to those above one may also show that the Lya-�� punov function candidate V (t, q, q) defined in (12.14) is bounded from aboveby�# �#T �# �#� �q q�Max{Kp} � �1 �Max{M}1��# �# �# �#� �V (t, q, q) d"q q2 � �� 1 �Max{M} �Max{M} �which is positive definite and radially unbounded since the condition2�Max{Kp} >�2�1 �Max{M}�� is trivially satisfied under hypothesis (12.13) on Kp.This means that V (t, q, q)is decrescent.Time DerivativeThe time derivative of the Lyapunov function candidate (12.14) along thetrajectories of the closed-loop system (12.7) is276 12 Feedforward Control and PD Control plus FeedforwardT 1 T� � � � � � � �� V (t, q, q) = q -Kpq - Kvq - C(q, q)q - h(t, q, q) + q @(q)q2T� � � �� + qT Kpq + �q Sech2(q)T M(q)q + �tanh(q)T @(q)q� � �� + �tanh(q)T -Kpq - Kvq - C(q, q)q - h(t, q, q).1Using Property 4.2 which establishes the skew-symmetry of @ - C and2� �@(q) = C(q, q) +C(q, q)T , the time derivative of the Lyapunov functioncandidate yieldsT T� � � � � �� V (t, q, q) =-q Kvq + �q Sech2(q)T M(q)q - �tanh(q)T Kpq� �� - �tanh(q)T Kvq + �tanh(q)T C(q, q)T qT� � �� - q h(t, q, q) - � tanh(q)T h(t, q, q).(12.15)� �� We now proceed to upper-bound V (t, q, q) by a negative definite function�� in terms of the states q and q.To that end, it is convenient to find upper-bounds for each term of (12.15).The first term of (12.15) may be trivially bounded byT� � �� -q Kvq d"-�min{Kv} q 2.(12.16)To upper-bound the second term of (12.15) we first recall that the�vectorial tangent hyperbolic function tanh(q) defined in (12.10) satisfiesSech (q)q d" �4 q2� �� with�4 > 0.From this, it follows thatTq 2� � �� q Sech2(q)T M(q)q d" ��4 �Max{M} �.On the other hand, notice that in view of the fact that Kp is a diagonal� � �positive definite matrix, and tanh(q) 2 d" �3 tanh(q)T q, we get� � �� 3 tanh(q)T Kpq e" ��min{Kp} tanh(q) 2which finally leads to the important inequality,�min{Kp}� � �-�tanh(q)T Kpq d"-� tanh(q) 2.�3�� A bound on �tanh(q)T Kvq is obtained straightforwardly and is given byq� �� tanh(q)T Kvq d" ��Max{Kv} � tanh(q).�� The upper-bound on the term �tanh(q)T C(q, q)T q must be carefully cho-sen.Notice that12.2 PD Control plus Feedforward 277T� �� tanh(q)T C(q, q)T q = �q C(q, q)tanh(q)q�� d" � � C(q, q)tanh(q).Considering again Property 4.2 but in its variant that establishes the existenceof a constant kC such that C(q, x)y d"kC x y for all q, x, y " IRn,1 1we have� �� tanh(q)T C(q, q)T q d" �kC q q tanh(q) ,1� �� d" �kC q qd - q tanh(q) ,1�� d" �kC q qd tanh(q)1� �� +�kC q q tanh(q).1� �Making use of the property that tanh(q) d"�2 for all q " IRn, we get� � �� tanh(q)T C(q, q)T q d" � kC qd M q tanh(q) + ��2 kC q 2.1 1At this point it is only left to find upper-bounds on the two terms which�contain h(t, q, q).This study is based on the use of the characteristics es-�� tablished in Property 4.4 on the vector of residual dynamics h(t, q, q), whichindicates the existence of constants kh1, kh2 e" 0 which may be computedby (4.24) and (4.25) such that the norm of the residual dynamics satisfies(4.15),h(t, q, q) d" kh1 q� �� + kh2 tanh(q).�T� �First, we study the term -q h(t, q, q):Tq h(t,� � � �-q h(t, q, q) d" q, q) ,� ��d" kh1 q 2 + kh2 q tanh(q).The remaining term satisfiesh(t, ��-�tanh(q)T h(t, q, q) d" � tanh(q) q, q) ,��d" � kh1 q tanh(q) + � kh2 tanh(q) 2.(12.17)� �� The bounds (12.16) (12.17) yield that the time derivative V (t, q, q) in(12.15), satisfies278 12 Feedforward Control and PD Control plus Feedforward�� V (t, q, q) d"�# �#T �min{Kp} 1 kh2- kh2 -a -� �tanh(q) tanh(q)�# �3�#� 2-��# �#1 kh2 1� �q q-a - [�min{Kv}-kh1] - b� 2 �R(�)(12.18)where1�a = [�Max{Kv} + kC qd M + kh1] ,12b = �4 �Max{M} + �2 kC.1According to the theorem of Sylvester, in order for the matrix R(�) to bepositive definite it is necessary and sufficient that the component R11 and thedeterminant det{R(�)} be strictly positive.With respect to the first conditionwe stress that the gain Kp must satisfy�min{Kp} e"�3 kh2.(12.19)On the other hand, the determinant of R(�) is given by1 �min{Kp}det{R(�)} = - kh2 [�min{Kv}-kh1]� �32�min{Kp} 1 kh2- - kh2 b - a +.�3 � 2The latter must be strictly positive for which it is necessary and sufficientthat the gain Kp satisfies[2 � a + kh2]2�min{Kp} >�3 + kh2 (12.20)4 � [�min{Kv}-kh1 - �b]while it is sufficient that Kv satisfies�min{Kv} >kh1 + � b (12.21)for the right-hand side of the inequality (12.20) to be positive.Observe thatin this case the inequality (12.19) is trivially implied by (12.20).Notice that the inequalities (12.21) and (12.20) correspond precisely tothose in (12.11) and (12.12) as the tuning guidelines for the controller.This� �� means that R(�) is positive definite and therefore, V (t, q, q) is globally nega-tive definite.According to the arguments above, given a positive constant � we maydetermine gains Kp and Kv according to (12.11) (12.13) in a way that the12.2 PD Control plus Feedforward 279�� function V (t, q, q) given by (12.14) is globally positive definite while V (t, q, q)� �expressed as (12.18) is globally negative definite.For this reason, V (t, q, q) is astrict Lyapunov function.Theorem 2.4 allows one to establish global uniformasymptotic stability of the origin of the closed-loop system.Tuning ProcedureThe stability analysis presented in previous sections allows one to obtain atuning procedure for the PD control law plus feedforward.This method deter-mines the smallest eigenvalues of the symmetric design matrices Kp and Kv with Kp diagonal which guarantee the achievement of the motion controlobjective.The tuning procedure may be summarized as follows." Derivation of the dynamic robot model to be controlled.Particularly, com-�putation of M(q), C(q, q) and g(q) in closed form [ Pobierz całość w formacie PDF ]