Abstract
In this work we compare equations of motion using the so-called inertial quasi-velocities. As a result of these velocities we obtain two first-order decoupled equations of motion instead of one second-order differential equation of motion. The methods presented here, solve in a way, the problem of nonlinear dynamic decoupling. The first and the second method result from diagonalized Lagrangian robot dynamics (Jain and Rodriguez, IEEE Trans Robot Autom 11:571–584, 1995) and are known as normalized and unnormalized quasi-velocities. The third method described by Junkins and Schaub (J Astronaut Sci 45:279–295, 1997) offers eigenfactor quasi-coordinate velocities formulation for multibody dynamics. As a consequence of using transformation given by Loduha and Ravani (Trans ASME J Appl Mech 62:216–222, 1995) we obtain decoupled equations of motion in terms of modified generalized velocity components. Here we limit all these methods to serial manipulators. The novelty of this paper consists in physical interpretation of the quasi-velocities and discussion concerning equations of motion, the kinetic energy shaping, relationship between each of them and properties useful for simulation and control purposes. Also forward dynamics algorithms and their computational complexity in terms of new velocities are given. Simulation results illustrate the theoretical investigations. We conclude that all methods offer interesting possibilities for dynamic simulation and future control investigations.
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Abbreviations
- \(\mathcal{N}\) :
-
number of joints and number of degrees of freedom
- \(\varvec{\theta},\varvec{\dot{\theta}},\varvec{\ddot{\theta}} \in R^{\mathcal{N}}\) :
-
vectors of generalized positions, velocities and accelerations, respectively
- \({\bf M}(\varvec{\theta}) \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
system mass matrix in classical equations of motion
- \({\bf G}(\varvec{\theta}) \in R^{\mathcal{N}}\) :
-
vector of gravitational forces in classical equations of motion
- Q \(\in R^{\mathcal{N}}\) :
-
vector of generalized forces in classical equations of motion
- \({\bf M}_{\rm p}(\varvec{\theta}) \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
system mass matrix in Poincare’s equations of motion
- \({\bf C}_{\rm p}(\varvec{\theta},{\bf p}) \in R^{\mathcal{N}}\) :
-
vector of Coriolis and centrifugal forces in Poincare’s equations of motion
- \({\bf G}_{\rm p}(\varvec{\theta}) \in R^{\mathcal{N}}\) :
-
vector of gravitational forces in Poincare’s equations of motion
- \({\bf p} \in R^{\mathcal{N}}\) :
-
vector of quasi-velocities in Poincare’s equations of motion
- \({\bf Q}_{\rm p} \in R^{\mathcal{N}}\) :
-
vector of generalized forces in Poincare’s equations of motion
- \({\bf v}_{\rm q},\mathbf{\dot{v}}_{\rm q} \in R^{\mathcal{N}}\) :
-
vector of kinematical quasi-velocities and its time derivative
- \(\varvec{\mathcal{A}}(\varvec{\theta}) \in R^{\mathcal{N} \times \mathcal{N}}\) :
-
dimensional configuration-dependent transformation matrix between quasi-velocities and joint velocities
- \({\bf B}_0 \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
dimensional configuration-dependent transformation matrix between joint velocities and quasi-velocities
- \({\bf b}_0 \in R^{\mathcal{N}}\) :
-
additional configuration-dependent vector arising from matrix B 0
- \(\varvec{\dot{q}} \in R^{\mathcal{N}}\) :
-
vector of time derivatives of quasi-coordinates
- \({\bf dq} \in R^{\mathcal{N}}\) :
-
vector of time differentials of quasi-coordinates
- \({\bf h}^{*}(\varvec{\theta},{\bf v}_{\rm q},t) \in R^{\mathcal{N}}\) :
-
vector which represents the sum of applied forces, gyroscopic terms, centrifugal forces and Coriolis effects
- \({\bf F}_r \in R^{\mathcal{N}}\) :
-
generalized active forces vector
- \({\bf F}_r^{*} \in R^{\mathcal{N}}\) :
-
generalized inertia forces vector
- \(\varvec{\nu} \in R^{\mathcal{N}}\) :
-
vector of normalized quasi-velocities
- \(\varvec{\xi} \in R^{\mathcal{N}}\) :
-
vector of unnormalized quasi-velocities
- \({\bf C}(\varvec{\theta},\varvec{\nu}) \in R^{\mathcal{N}}\) :
-
vector of Coriolis and centrifugal forces in diagonalized normalized equations of motion
- \({\bf C}(\varvec{\theta},\varvec{\xi}) \in R^{\mathcal{N}}\) :
-
vector of Coriolis and centrifugal forces in diagonalized unnormalized equations of motion
- \({\bf G}_{\nu}(\varvec{\theta}) = {\bf m}^{-1}(\varvec{\theta}){\bf G}(\varvec{\theta}) \in R^{\mathcal{N}}\) :
-
vector of gravitational forces in diagonalized normalized equations of motion
- \({\bf D=HPH}^{{\rm T}} \in R^{\mathcal{N} \times \mathcal{N}}\) :
-
articulated inertia about joint axes matrix
- \({\bf H} \in R^{\mathcal{N}\times 6\mathcal{N}}\) :
-
projection operator for all joint axes
- \({\bf P} \in R^{6 \mathcal{N} \times 6 \mathcal{N}}\) :
-
articulated inertia matrix
- \({\bf G}_{\xi}(\varvec{\theta})={\bf D}^{{1/2}}{\bf m}^{-1}(\varvec{\theta}){\bf G}(\varvec{\theta}) \in R^{\mathcal{N}}\) :
-
vector of gravitational forces in diagonalized unnormalized equations of motion
- \({\bf M}(\varvec{\theta}) \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
spatial operator – “square root” of mass matrix \({\bf M}(\varvec{\theta})\), namely \({\bf M}(\varvec{\theta}) = {\bf m}(\varvec{\theta}){\bf m}^{{\rm T}}(\varvec{\theta})\) which is expressed as \({\bf m}(\varvec{\theta}) = [ {\bf I}\,{+}\,{\bf H}\varvec{\phi} {\bf K}]{\bf D}^{{1/2}}\)
- \(\varvec{\phi} \in R^{6 \mathcal{N}\times 6 \mathcal{N}}\) :
-
rigid manipulator force transformation matrix
- \({\bf K} \in R^{6 \mathcal{N}\times \mathcal{N}}\) :
-
shifted Kalman gain matrix
- \(\varvec{\dot{\rm m}}(\varvec{\theta}) \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
time derivative of factor \({\bf M}(\varvec{\theta})\)
- \(\varvec{\epsilon} \in R^{\mathcal{N}}\) :
-
vector of normalized quasi-moments
- \(\varvec{\kappa} \in R^{\mathcal{N}}\) :
-
vector of unnormalized quasi-moments
- \(\mathcal{O}_k\) :
-
origin of the frame attached to the k-th link
- \({\theta}_k,\dot{\theta}_k,\ddot{\theta}_k\)kth:
-
generalized position, velocity and acceleration, respectively
- lk,j ∈R3:
-
vector from origin \(\mathcal{O}_k\) to origin \(\mathcal{O}_j\)
- \(\varvec{\phi}_{k,k-1} \in R^{6\times 6}\) :
-
operator which transforms rigid quantities from the kth to the (k − 1)th joint defined as:
$$\varvec{\phi}_{k,k-1} = \left[\begin{array}{ll}{\bf I}& \tilde{{\bf l}}_{k,k-1}\\ {\bf 0}& {\bf I} {\tilde{{\bf l}}}\end{array}\right] $$and is a skew symmetric matrix
$$\tilde{{\bf l}}=\left[\begin{array}{lll} 0& -l_{z} & l_y\\ l_{z} & 0 & -l_{x}\\ -l_{y} & l_{x} & 0 \end{array}\right]$$where l x , l y , l z are elements x, y and z of vectorl, 0 and I denote zero and unit matrices of appropriate dimensions
- k+1R k ∈R3 × 3:
-
direction cosine matrix between \(\mathcal{O}_k\) and \(\mathcal{O}_{k+1}\) according to the modified Denavit–Hartenberg notation [14]
- \(\varvec{\psi}_{k,k-1} \in R^{6\times 6}\) :
-
operator which transforms articulated quantities from the kth to the (k − 1)th joint
- α k :
-
angle for the kth joint in the modified Denavit–Hartenberg notation [14]
- A k ∈R6 × 6:
-
spatial rotation matrix defined as:
$${\bf A}_k = \left[\begin{array}{ll}^{k+1}{\bf R}_k & {\bf 0}\\ {\bf 0} & ^{k+1}{\bf R}_k\end{array}\right]$$ - (.)T :
-
transpose operation
- \({\bf h}^{\rm T}_k \in R^3\) :
-
axis of rotation or axis of translation for the kth joint
- \({\bf H}^{\rm T}_{k} \in R^6\) :
-
joint map matrix for the kth joint:
$$\begin{array}{lll}{\bf H}^{{\rm T}}_k= & \left[ {\bf h}_k, \; {\bf 0}^{{\rm T}}\right]^{{\rm T}} & {\rm rotational \; joint} \\ {\bf H}^{{\rm T}}_k= & \left[{\bf 0}^{{\rm T}}, \;{\bf h}_k\right]^{{\rm T}} & {\rm translational \; joint}\end{array}$$ - m k :
-
mass of the kth link
- p k ∈R3:
-
vector from \(\mathcal{O}_k\) to the kth link’s center mass
- \({\varvec{\mathcal{I}}}_k \in R^{3\times 3}\) :
-
inertia tensor of kth link with respect to \(\mathcal{O}_k\)
- M k ∈R6 × 6:
-
spatial inertia matrix of the kth link expressed in the coordinate \(\mathcal{O}_k\) defined as:
$${\bf M}_k=\left[ \begin{array}{ll} \varvec{\mathcal{I}}_k & {\bf m}_k\tilde{{\bf p}}_k\\ -{\bf m}_k \tilde{{\bf p}}_k &{\bf m}_{k}{\bf I} \end{array} \right] $$ - V k ∈R6:
-
spatial velocity vector of the kth body frame,
$${\bf V}_k=\left[ \begin{array}{l} \varvec{\omega}_k\\ {\bf v}_k \end{array} \right]$$ - \(\varvec{\omega}_{k} \in R^{3}\) :
-
angular velocity of the kth body
- v k ∈R3:
-
linear velocity of the kth body
- n k ∈R6:
-
spatial bias acceleration vector for the kth link: for rotational joint or translational joint defined as [36, 37]
$${\bf n}_k=\left[ \begin{array}{ll}\mbox{}^{k+1}{\bf R}_k (\varvec{\omega}_k &\times {\bf h}_k \dot{\theta}_k)\\ \mbox{}^{k+1}{\bf R}_k ({\bf v}_k &\times {\bf h}_k \dot{\theta}_k)\end{array}\right]$$and for translational joint [36, 37]
$${\bf n}_k=\left[\begin{array}{l} \mbox{}{\bf 0}\\ \mbox{}^{k+1}{\bf R}_k (\varvec{\omega}_k \times {\bf h}_k \dot{\theta}_k) \end{array}\right] $$ - G ak ∈R6:
-
Kalman gain vector for the kth joint
- P k ∈R6 × 6:
-
spatial articulated inertia matrix for the kth joint
- D k :
-
articulated inertia about the kth joint axis
- ν k :
-
kth normalized quasi-velocity
- \(\dot{\nu}_k\) :
-
kth normalized quasi-acceleration
- ξ k :
-
kth unnormalized quasi-velocity
- \(\dot{\xi}_k\) :
-
kth unnormalized quasi-acceleration
- ε k :
-
kth normalized quasi-moment
- C ν k :
-
Coriolis term for the kth joint in normalized diagonalized equations of motion
- G ν k :
-
gravity term for the kth joint in normalized diagonalized equations of motion
- κ k :
-
kth unnormalized quasi-moment
- C ξ k :
-
Coriolis term for the kth joint in unnormalized diagonalized equations of motion
- G ξ k :
-
gravity term for the kth joint in unnormalized diagonalized equations of motion
- b k ∈R6:
-
spatial bias forces vector calculated as (both for rotational and translational joints) [36, 37]:
$${\bf b}_k=\left[ \begin{array}{l} \varvec{\omega}_k \times \varvec{\mathcal{I}}_k \varvec{\omega}_k + m_k {\bf p}_k \times [\varvec{\omega}_k \times {\bf v}_k]\\ m_k [ \varvec{\omega}_k \times {\bf v}_k +\varvec{\omega}_k \times (\varvec{\omega}_k \times {\bf p}_k)] \end{array} \right]$$ - G k ∈R3:
-
three-dimensional gravitational forces vector
- \({\bf b}_{{\rm g}k}=(-{\bf p}_k \times {\bf G}_k, -{\bf G}_k)^{{\rm T}} \in R^6\) :
-
spatial gravitational forces vector
- \(\varvec{\tau} \in R^{\mathcal{N}}\) :
-
generalized force acting at the manipulator
- \(\varvec{\eta} \in R^{\mathcal{N}}\) :
-
eigenfactor quasi-coordinate velocity (EQV) vector
- \(\varvec{\dot{\eta}} \in R^{\mathcal{N}}\) :
-
eigenfactor quasi-coordinate acceleration vector
- \({\bf SC}_e \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
EQV rate transformation matrix
- \({\bf D}_{\rm e}=\hbox{diag}(\lambda_i) \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
diagonal matrix containing eigenvalues of the mass matrix \({\bf M}(\varvec{\theta})\)
- λ i :
-
ith eigenvalue of the mass matrix \({\bf M}(\varvec{\theta})\)
- \(\dot{\lambda}_i\) :
-
time derivative of the eigenvalue λ i
- \({\bf E}={\bf C}_{\rm e}^{\rm T} \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
orthogonal real matrix of eigenvectors of \({\bf M}(\varvec{\theta})\)
- \(\varvec{\Omega}_{\rm e} \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
skew symmetric matrix in which each element Ω ij represents a generalized eigenvector axis angular velocity
- \({\bf M}_{\theta}=\partial{{\bf M}}/\partial{\varvec{\theta}}\) :
-
partial derivative with respect to the configuration vector \(\varvec{\theta}\)
- \({\bf C}(\varvec{\theta},\varvec{\dot{\theta}},\varvec{\eta}) \in R^{\mathcal{N}}\) :
-
vector of Coriolis and centrifugal forces in EQV formulation
- \(\varvec{\varepsilon} \in R^{\mathcal{N}}\) :
-
vector of quasi-moments in EQV formulation
- \({\bf u} \in R^{\mathcal{N}}\) :
-
generalized velocity component (GVC) vector
- \(\dot{{\bf u}} \in R^{\mathcal{N}}\) :
-
time derivative of the GVC vector
- \(\varvec{\Upsilon} \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
rate transformation matrix in GVC formulation which depends on mass matrix of the system and kinematical parameters
- \(\varvec{\dot{\Upsilon}} \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
time derivative of the matrix \(\varvec{\Upsilon}\)
- \({\bf N} \in R^{\mathcal{N}\times \mathcal{N}}\) :
-
diagonal mass matrix in GVC formulation
- \({\bf C}\left(\varvec{\theta},\varvec{\dot{\theta}},{\bf u}\right) \in R^{\mathcal{N}}\) :
-
vector of Coriolis and centrifugal forces in GVC equations of motion
- \(\varvec{\pi} \in R^{\mathcal{N}}\) :
-
vector of quasi-moments in GVC formulation
- \({\bf J}_{k} \in R^{3\times \mathcal{N}}\) :
-
partial derivative of the kth body mass center position with respect to the inertial reference frame
- \(\varvec{\Omega}_{k} \in R^{3\times \mathcal{N}}\) :
-
partial derivative of the kth body angular velocity with respect to the time derivative of the generalized coordinate vector
- \({\bf W}_{k}=\varvec{\tilde{\omega}}_{k} \in R^{3\times 3}\) :
-
angular velocity matrix associated with the kth body written in terms of the kth body
- f k ∈R3 where \({\bf f}_{k}={\bf f}_{{\rm g}k}+{\bf f}_{{\rm e}k}\):
-
resultant active force acting at the mass center of the kth body, where \({\bf f}_{{\rm g}k}\) and \({\bf f}_{{\rm e}k}\) denote gravitational and external forces, respectively
- \(\varvec{\tau}_{{\rm R}k} \in R^3\) :
-
where \(\varvec{\tau}_{{\rm R}k}=\varvec{\tau}_{{\rm g}k}+\varvec{\tau}_{{\rm e}k}\) resultant moment of the kth body, where \(\varvec{\tau}_{{\rm g}k}\) and \(\varvec{\tau}_{{\rm e}k}\) denote gravitational and external moments, respectively
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Herman, P., Kozłowski, K. A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators. Arch Appl Mech 76, 579–614 (2006). https://doi.org/10.1007/s00419-006-0021-0
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DOI: https://doi.org/10.1007/s00419-006-0021-0