MEZA et al.: FUZZY SELF-TUNING <strong>PID</strong> SEMIGLOBAL REGULATOR FOR ROBOT MANIPULATORS 2711matrices as functions of the robot configuration. This leads tothe following proposed control law:∫ tτ = K p (˜q)˜q − K v (˜q) ˙q + K i (ω) ˜q(σ)dσ (8)where K p (˜q),K v (˜q), and K i (ω) are positive definite diagonaln × n matrices, whose entries are denoted by k pi (˜q i ), k vi (˜q i ),and k ii (ω i ), respectively, ˜q = q d − q denotes the position errorvector, and ˙ω = α˜q − ˙q, with α>0.For stability analysis purposes, the control law (8) can berewritten as [31], [33]τ =K ′ p(˜q)˜q−K v (˜q) ˙q+K ′ i(ω)∫ t00(α˜q(σ)+ ˙˜q(σ))dσ (9)where K ′ p(˜q) =K p (˜q) − (K i (ω)/α) and K ′ i (ω) =(K i (ω)/α), with α>(λ M {K i }/λ m {K p }).This latter condition ensures that K ′ p(˜q) > 0.Theα constantis introduced in order to make easier the stability analysis, andthis will be used as a parameter of the Lyapunov function.Assumption 1: There exist positive constants k pli , k pui , k vli ,k vui , k ili , and k iui such that Lemma 1 can be applied. That is∫˜q12 ˜qT K pu˜q ′ ≥0∫˜q12 ˜qT K vu˜q ≥12 ωT K ′ iuω ≥0∫ ω0ξK p(ξ)dξ ′ ≥ 1 2 ˜qT ′K pl˜q (10)ξK v (ξ)dξ ≥ 1 2 ˜qT K vl˜q (11)ξK ′ i(ξ)dξ ≥ 1 2 ωT K ′ ilω (12)where K pu, ′ Kpl ′ , K vu, K vl , K iu ′ , and K′ ilare n × n constantpositive definite diagonal matrices whose entries arek pu ′ i,kpl ′ i,k vui ,k vli ,k iu ′ i, and kil ′ i, respectively, with i =1, 2,...,n.Assumption 2: In an ɛ-neighborhood N(ω,ɛ)={ω ∈ IR n :‖ω‖ n∑j=1and α is chosen in such a way that it satisfiesk vli∑ nj=1 max q |M ij (q)| >α>max q∣ ∣∣∣ ∂g i (q)∂q j∣ ∣∣∣(16)k ilik pli − ∑ nj=1 max q |∂g i (q)/∂q j | .(17)B. Time Derivative of the Lyapunov Function CandidateThe time derivative of the Lyapunov function candidate (15)along the trajectories of the closed-loop equation (13) is˙V (˜q, ˙q, ω) =− ˙q T [K v (˜q) − αM(q)] ˙q − α˜q T C(q, ˙q) T ˙q−α˜q T [K p (˜q) − K i (ω)/α] ˜q − α˜q T [g(q d ) − g(q)]where we have used the Leibnitz’ rule <strong>for</strong> differentiation ofintegrals and Property 1. Again, following similar steps given
Change language