Velocity tracking control algorithm in terms of quasi-velocities Przemyslaw Herman * Krzysztof Kozlowski ** * Chair of Control and Systems Engineering, Poznan University of Technology, Poznan, Poland (e-mail: przemyslaw.herman@ put.poznan.pl). ** Chair of Control and Systems Engineering, Poznan University of Technology, Poznan, Poland (e-mail: krzysztof.kozlowski@ put.poznan.pl). Abstract: In this paper a tracking controller expressed in terms of the quasi-velocities for rigid manipulators described by the Poincar´ e equations in a matrix form is proposed. The quasi-velocities introduced by Hurtado (2004) are based on Cholesky decomposition of the system inertia matrix. Using the proposed, exponentially convergent, controller it is possible to recognize some dynamic features of the system and at the same time to ensure the velocity control. It is shown that the presented approach can be helpful for reduction of nonlinearities existing in the manipulator in its design phase. The control strategy was tested in simulation on a 3 d.o.f. spatial manipulator. 1. NOMENCLATURE N - number of degrees of freedom, θ, ˙ θ, ¨ θ ∈ R N - vectors of generalized positions, velocities, and accelerations, respectively, M (θ) ∈ R N ×N - system mass matrix, C(θ, ˙ θ)= ˙ M ˙ θ − 1 2 ˙ θ T M θ ˙ θ ∈ R N - vector of Coriolis and centrifugal forces in classical equations of motion, where the expression ˙ θ T M θ ˙ θ is the column vector col( ˙ θ T M θ k ˙ θ), M θ k = ∂M ∂θ k denotes the partial derivative of the inertia matrix M (θ) and ˙ M is its time derivative Koditschek [1985], G(θ) ∈ R N - vector of gravitational forces in classical equations of motion, Q ∈ R N - vector of generalized forces, f ( ˙ θ)= F v ˙ θ ∈ R N - vector of viscous friction forces, where F v means a diagonal positive definite matrix, ω ∈ R N - vector of quasi-velocities, C ω (θ, ν ) ∈ R N - vector of Coriolis and centrifugal forces in equations of motion expressed in terms of Poincar´ e equations of motion, G ω (θ) ∈ R N - vector of gravitational forces in terms of Poincar´ e equations of motion, f (θ, ˙ θ) ∈ R N - vector of friction forces in terms of Poincar´ e equations of motion, π ∈ R N - vector of quasi-moments terms of Poincar´ e equations of motion, (.) T transpose operation. 2. INTRODUCTION The tracking control problem of rigid manipulators is well known from the robotic literature (e.g. Canudas de Wit, Siciliano, Bastin [1996], Sciavicco and Siciliano [1996]). It concerns systems which are described by complex second- order nonlinear differential equations of motion. The in- verse dynamics controller leads to a decoupled system. There exist some developments concerning simplification of the robot dynamics formulation. One on them relies on introducing a canonical transformation. However, the necessary and sufficient conditions which ensure obtaining tracking of the canonical variables are very restrictive and rarely satisfied in practice Jain and Rodriguez [1995], Spong [1992]. Different approach, which is easier to realize, relies on diagonalization of the inertia matrix using quasi- velocities. There exist several methods resulting to first- order equations of motion with the identity inertia matrix Hurtado [2004], Jain and Rodriguez [1995], Junkins and Schaub [1997], Sovinsky et al. [2005]. Schaub and Junk- ins gave simulation results Schaub, J.L. Junkins [1997] concerning feedback control in terms of quasi-velocities defined in Junkins and Schaub [1997]. Moreover, two PD controllers realized using quasi-velocities Jain and Ro- driguez [1995], Junkins and Schaub [1997] were compared in reference Herman and Kozlowski [2001]. The aim of this paper is to show a controller which enables velocity trajectory tracking using quasi-velocities (QV) in- troduced by Hurtado [2004]. It is shown that the proposed controller guarantees exponential convergence of quasi- velocities error. The proposed control algorithm is based on the equations of motion with the identity inertia matrix. This fact implies that we obtain further insight into the manipulator dynamics. Additionally, an idea of desired quasi-velocity trajectory was explained. The QV controller can be helpful for evaluation manipulator dynamics be- cause it contains all quantities arising from decomposition of the system inertia matrix. Therefore, it gives some infor- mation which is necessary at the design step of manipula- tors in which simulation investigation plays an important role. We point at some advantages which are observable if the QV tracking controller is used. The control scheme was tested on a 3 d.o.f. spatial manipulators.