Dynamic modeling of parallel manipulators based on Lagrange–D’Alembert formulation and Jacobian/Hessian matrices

Abstract

This paper describes the development of a dynamic model for parallel manipulators based on the Lagrange–D’Alembert equation, the Hessian matrix of the kinetic energy of the manipulator, and the Jacobian/Hessian matrices of the constraint equations. The model generates the forward and inverse dynamics formulations of parallel manipulators. First, the kinematic relations between the input actuated variables and the output variables are defined in terms of the Jacobian and Hessian matrices of the constraint equations. Then, a new form of Lagrange–D’Alembert equation is introduced and used to derive the dynamic model of parallel manipulators. In the present model, the inertia, centrifugal, and Coriolis generalized forces/torques are obtained directly from Hessian matrix of the kinetic energy of the manipulator. The developed model is simple, straightforward, and appropriate for parallel processing techniques as the elements of the Jacobian and Hessian matrices can be calculated simultaneously. The developed model is validated using the conservation of work and energy principle. A trajectory tracking control of a simple 3RRR planar parallel manipulator is used to illustrate the developed model.

Appendices

Appendix A

The first constraint is

$$ h_{1} = \bigl[ x - x_{B1} - a \cos \theta _{1} - l \cos ( \alpha _{1} + \varphi ) \bigr]^{2} + \bigl[ y - y_{B1} - a \sin \theta _{1} - l \sin ( \alpha _{1} + \varphi ) \bigr] ^{2} - b^{2} =0. $$

The Hessian matrix of the first constraint equation is written as follows:

$$\begin{aligned} \boldsymbol{H}_{f_{1}} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \frac{\partial ^{2} h_{1}}{\partial \theta _{1}^{2}} & \cdots & \frac{\partial ^{2} h_{1}}{\partial \theta _{1} \partial \varphi }\\ \vdots & \ddots & \vdots \\ \frac{\partial ^{2} h_{1}}{\partial \varphi \partial \theta _{1}} & \cdots & \frac{\partial ^{2} h_{1}}{\partial \varphi ^{2}} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} h_{111} & \cdots & h_{11 \varphi }\\ \vdots & \ddots & \vdots \\ h_{1 \varphi 1} & \cdots & h_{1 \varphi \varphi } \end{array}\displaystyle \right ] \in \mathbb{R}^{6\times 6} \end{aligned}$$

where $\boldsymbol{H}_{f_{1}} \in \mathbb{R}^{6\times 6}$ is the Hessian matrix for the first constraint equation with respect to the generalized coordinates, $\boldsymbol{q} = [ \theta _{1} \ \theta _{2} \ \theta _{3} \ x \ y \ \varphi ]^{T}$, and its elements are:

$$\begin{aligned} h_{111} &= \frac{\partial ^{2} h_{1}}{\partial \theta _{1}^{2}} =2 a \bigl[ ( x - x_{B 1} ) \cos \theta _{1} + ( y - y _{B 1} ) \sin \theta _{1} - l \cos ( \alpha _{1} + \varphi - \theta _{1} ) \bigr] \\ h_{112} &= \frac{\partial ^{2} h_{1}}{\partial \theta _{1} \partial \theta _{2}} =0, \qquad h_{113} = \frac{\partial ^{2} h_{1}}{\partial \theta _{1} \partial \theta _{3}} =0, \qquad h_{11 x} = \frac{\partial ^{2} h_{1}}{\partial \theta _{1} \partial x} =2 a x \sin \theta _{1} \\ h_{11 y}& = \frac{\partial ^{2} h_{1}}{\partial \theta _{1} \partial y} =-2 a \cos \theta _{1}, \qquad h_{11 \varphi } = \frac{\partial ^{2} h_{1}}{\partial \theta _{1} \partial \varphi } =2 a l \cos ( \alpha _{1} + \varphi - \theta _{1} ) \end{aligned}$$

The first constraint is not a function of $\theta _{2}$ and $\theta _{3}$, thus

$$\begin{aligned} h_{12 i} &=0\quad \mbox{and}\quad h_{13 i} =0\quad (i=1, 2, \dots , 6) \\ h_{1 xx} &= \frac{\partial ^{2} h_{1}}{\partial x^{2}} =2, \qquad h_{1 xy} = \frac{\partial ^{2} h_{1}}{\partial x \partial y} =0, \qquad h_{1 x\varphi } = \frac{\partial ^{2} h_{1}}{\partial x \partial \varphi } =2 l \sin ( \alpha _{1} + \varphi ) \\ h_{1 yy} &= \frac{\partial ^{2} h_{1}}{\partial \varphi ^{2}} =2, \qquad h_{1 y\varphi } = \frac{\partial ^{2} h_{1}}{\partial y \partial \varphi } =-2 l \cos ( \alpha _{1} + \varphi ) \\ h_{1 \varphi \varphi } &= \frac{\partial ^{2} h_{1}}{\partial \varphi ^{2}} =2 ( y - y_{B 1} ) l \sin ( \alpha _{1} + \varphi ) +2 ( x - x_{B 1} ) l \cos ( \alpha _{1} + \varphi ) -2 a l \cos ( \alpha _{1} + \varphi - \theta _{1} ) \end{aligned}$$

Appendix B

The dynamic equations of the manipulator are developed by applying the Lagrangian equations of the first type. The Lagrangian equations of the first type are written in terms of a set of redundant coordinates. Therefore, the formulation requires a set of constraint equations derived from the kinematics of the manipulator [16]. The Lagrangian equations of the first type can be written as

$$\begin{gathered} \sum_{i=1}^{k} \lambda _{i} \frac{\partial h}{\partial q_{j}} = \frac{d}{dt} \biggl( \frac{\partial L}{\partial \dot{q}_{j}} \biggr) - \frac{\partial L}{\partial q_{j}} - \hat{Q}_{j} \quad j = 1\mbox{ to }k \end{gathered}$$

(B.1)

$$\begin{gathered} Q_{j} = \frac{d}{dt} \biggl( \frac{\partial L}{\partial \dot{q}_{j}} \biggr) - \frac{\partial L}{\partial q_{j}} - \sum_{i=1}^{k} \lambda _{i} \frac{\partial h_{i}}{\partial q_{j}}\quad j = k +1\mbox{ to }n \end{gathered}$$

(B.2)

where $L$ is the Lagrangian function, $L=T-V$, $T$ is the kinetic energy of the manipulator, and $V$ is the manipulator potential energy. $h _{i}$ denotes the $i$th constraint function, $k$ is the number of constraint functions, and $\lambda _{i}$ is the Lagrange multiplier; $q_{j}$ is the redundant coordinate. The number of redundant coordinates $n$ exceeds the number of degrees of freedom by $k$; $\hat{Q}_{j}$ represents the generalized forces contributed by externally applied forces; $Q_{j}$ is the actuator torque. The present manipulator is planar; therefore, the potential energy is zero and the Lagrangian function $L=T$ and is defined by Eq. (57).

Calculating the different terms of Eqs. (B.1) and (B.2), we obtain a system of dynamical equations. For $j=1, 2,\ \mbox{and} \ 3$, they are:

$$\begin{aligned} &2 \sum_{i=1}^{3} \lambda _{i} \bigl( x- x_{Bi} -a \cos \theta _{i} - l_{i} \cos ( \alpha _{i} +\varphi ) \bigr) \\ &\quad = ( m_{b} + m_{p} ) \ddot{x} \\ &\qquad{} + \frac{1}{3} m_{b} \sum_{i=1}^{3} \bigl( l_{i} \sin ( \alpha _{i} +\varphi ) \ddot{\varphi } + l_{i} \cos ( \alpha _{i} +\varphi ) \dot{\varphi }^{2} \bigr) \\ &\qquad{} - \frac{1}{6} a\ m_{b} \sum _{i=1}^{3} \bigl( \sin \theta _{i} \ddot{\theta }_{i} + \cos \theta _{i} \dot{\theta }_{i}^{2} \bigr) - F_{Px} \end{aligned}$$

(B.3)

$$\begin{aligned} &2 \sum_{i=1}^{3} \lambda _{i} \bigl( y- y_{Bi} -a \sin \theta _{i} - l_{i} \sin ( \alpha _{i} +\varphi ) \bigr) \\ &\quad = ( m_{b} + m_{p} ) \ddot{y} \\ &\qquad{} - \frac{1}{3} m_{b} \Biggl( \sum_{i=1}^{3} l_{i} \cos ( \alpha _{i} +\varphi ) \ddot{\varphi } - l_{i} \sin ( \alpha _{i} +\varphi ) \dot{\varphi }^{2} \Biggr) \\ &\qquad{} - \frac{1}{6} a m_{b} \Biggl( \sum_{i=1}^{3} \cos \theta _{i} \ddot{\theta }_{i} - \sin \theta _{i} \dot{\theta }_{i}^{2} \Biggr) - F_{Py} \end{aligned}$$

(B.4)

$$\begin{aligned} &2 l \sum_{i=1}^{3} \lambda _{i} \bigl( \ \bigl[ ( x- x_{Bi} ) \sin ( \alpha _{i} +\varphi ) - ( y- y_{Bi} ) \cos ( \alpha _{i} +\varphi ) -a \sin ( \alpha _{i} +\varphi - \theta _{i} ) \bigr] \bigr) \\ &\quad = \biggl[ \frac{1}{3} \bigl( l_{1}^{2} + l_{2}^{2} + l_{3}^{2} \bigr) m_{b} + I_{p} \biggr] \ddot{\varphi } \\ &\qquad{} + \frac{1}{3} m_{b} \Biggl( \sum _{i=1}^{3} l_{i} \sin ( \alpha _{i} +\varphi ) \ddot{x} - l_{i} \cos ( \alpha _{i} +\varphi ) \ddot{y} \Biggr) \\ &\qquad{} - \frac{1}{6} a m_{b} \sum_{i=1}^{3} \cos ( \alpha _{i} +\varphi - \theta _{i} ) \ddot{\theta }_{i} - \tau _{P} \end{aligned}$$

(B.5)

where $F _{px}$, $F _{py}$, and $\tau _{p}$ are the components of the applied forces and torques on the moving platform.

For $j = 4$, 5, and 6, we have

$$\begin{aligned} \tau _{1} &= \frac{1}{3} ( m_{a} + m_{b} ) a^{2} \ddot{\theta }_{1} \\ &\quad{} - \frac{1}{6} a m_{b} \bigl[ \ddot{x} \sin \theta _{1} - \ddot{y} \cos \theta _{1} + l_{1} \cos ( \alpha _{1} +\varphi - \theta _{1} ) \ddot{\varphi } \bigr] \\ &\quad{} + \frac{1}{6} a m_{b} l_{1} \sin ( \alpha _{1} +\varphi - \theta _{1} ) \dot{\varphi }^{2} \\ &\quad{} -2 \lambda _{1} a \bigl[ ( x- x_{B1} ) \sin \theta _{1} - ( y- y_{B1} ) \cos \theta _{1} + l_{i} \sin ( \alpha _{1} +\varphi - \theta _{1} ) \bigr] \end{aligned}$$

(B.6)

$$\begin{aligned} \tau _{2} & = \frac{1}{3} ( m_{a} + m_{b} ) a^{2} \ddot{\theta }_{2} \\ &\quad{} - \frac{1}{6} a m_{b} \bigl[ \ddot{x} \sin \theta _{2} - \ddot{y} \cos \theta _{2} + l_{2} \cos ( \alpha _{2} +\varphi - \theta _{2} ) \ddot{\varphi } \bigr] \\ &\quad{} + \frac{1}{6} a m_{b} l_{2} \sin ( \alpha _{2} +\varphi - \theta _{2} ) \dot{\varphi }^{2} \\ &\quad{} - 2 \lambda _{2} a \bigl[ ( x- x_{B2} ) \sin \theta _{2} - ( y- y_{B2} ) \cos \theta _{2} + l_{i} \sin ( \alpha _{2} +\varphi - \theta _{2} ) \bigr] \end{aligned}$$

(B.7)

$$\begin{aligned} \tau _{3} &= \frac{1}{3} ( m_{a} + m_{b} ) a^{2} \ddot{\theta }_{3} \\ &\quad{} - \frac{1}{6} a m_{b} \bigl[ \ddot{x} \sin \theta _{3} - \ddot{y} \cos \theta _{3} + l_{3} \cos ( \alpha _{3} +\varphi - \theta _{3} ) \ddot{\varphi } \bigr] \\ &\quad{} + \frac{1}{6} a m_{b} l_{3} \sin ( \alpha _{3} +\varphi - \theta _{3} ) \dot{\varphi }^{2} \\ &\quad{} - 2 \lambda _{3} a \bigl[ ( x- x_{B3} ) \sin \theta _{3} - ( y- y_{B3} ) \cos \theta _{3} + l_{3} \sin ( \alpha _{3} +\varphi - \theta _{3} ) \bigr] \end{aligned}$$

(B.8)

These two sets of equations, Eqs. (B.3)–(B.5) and (B.6)–(B.8), represent the dynamic model of the parallel manipulator. Equations (B.3)–(B.5) form a set of linear equations in three unknowns, from which the three Lagrange multipliers can be determined. Once Lagrange multipliers are found, the actuator torques are determined from the second set of Eqs. (B.6)–(B.8).