Abstract
In this paper a tracking controller expressed in terms of the normalized quasi-velocities (NQV) for rigid manipulators is proposed. These quasi-velocities introduced by [Jain and Rodriguez, IEEE Trans. Robot. Autom., 11:571–584, 1995] are utilized here in order to reveal some useful features which are observable if we track a desired quasi-velocity trajectory. The presented controller in terms of NQV is exponentially convergent. Moreover, some geometrical interpretation of the normalized quasi-variables based on Riemannian geometry is given. It is shown that the controller can be helpful for evaluation and reduction of dynamical couplings existing in the system. As a result it is helpful at the design step of manipulators. The control strategy was tested in simulation on two 3 d.o.f. spatial manipulators.
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Abbreviations
- \({\cal N}\) :
-
number of degrees of freedom
- \({{\theta}},\dot{{\theta}},\ddot{{\theta}} \in R^{\cal N}\) :
-
vectors of generalized positions, velocities, and accelerations, respectively
- \({{M}}({\theta}) \in R^{{\cal N}\times {\cal N}}\) :
-
system mass matrix
- \(C({\theta},\dot{{\theta}})=\dot{{M}}\dot{{\theta}}-\frac{1}{2}\dot{{\theta}}^T {M}_{{\theta}}\dot{{\theta}} \in R^{\cal N}\) :
-
vector of Coriolis and centrifugal forces in classical equations of motion, where the expression \(\dot{{\theta}}^T {M}_{{\theta}}\dot{{\theta}}\) is the column vector \({col}(\dot{{\theta}}^T {M}_{{\theta}_k}\dot{{\theta}}), {M}_{{\theta}_k}=\frac{\partial {M}}{\partial {\theta}_k}\) denotes the partial derivative of the mass matrix \({{M}}({\theta})\) and \(\dot{{M}}\) is its time derivative [9]
- \(G({\theta}) \in R^{\cal N}\) :
-
vector of gravitational forces in classical equations of motion
- \({\tau} \in R^{\cal N}\) :
-
vector of generalized forces
- \({\nu} \in R^{\cal N}\) :
-
vector of normalized quasi-velocities (NQV)
- \({H} \in R^{{\cal N}\times 6{\cal N}}\) :
-
projection operator for all joint axes
- \({{P}} \in R^{6{\cal N}\times 6{\cal N}}\) :
-
articulated inertia matrix
- \({D}={H}{P}{H}^{T} \in R^{{\cal N}\times {\cal N}}\) :
-
articulated inertia about joint axes, a diagonal matrix \({{D}}={D}({\theta})\)
- \(C_{\nu}({{\theta}},{{\nu}})=m^{-1}(\theta)C(\theta,\dot{\theta})-\dot{m}^{T}(\theta)\dot{\theta} \in R^{\cal N}\) :
-
vector of Coriolis and centrifugal forces in equations of motion expressed in terms of NQV vector
- \(G_{\nu}({\theta})=m^{-1}(\theta)G(\theta) \in {R}^{\cal N}\) :
-
vector of gravitational forces in equations of motion expressed in terms of NQV vector
- \({m}({{\theta}}) \in R^{{\cal N}\times {\cal N}}\) :
-
spatial operator "square root" of mass \(M({\theta})\), namely \(M({\theta})=m({\theta}){{{m}}}^{{T}}({\theta})\)
- \(\dot{{m}}({{\theta}}) \in R^{{\cal N}\times {\cal N}}\) :
-
time derivative of factor \({m}({{\theta}})\)
- \({\epsilon}=m^{-1}(\theta)\tau \in {R}^{\cal N}\) :
-
vector of normalized quasi-moments,
- \((.)^{T}\) :
-
transpose operation
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Herman, P. Tracking Controller Using Normalized Quasi-velocities. J Intell Robot Syst 47, 87–100 (2006). https://doi.org/10.1007/s10846-006-9073-1
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DOI: https://doi.org/10.1007/s10846-006-9073-1