Kinematic and Dynamic Model of a Simplified 3-DOF SPARA Robot

1. Kinematics

Structure

Degrees of Freedom: Pure translational motion (x, y, z).
Actuators: 3 legs (Legs 1–3), with redundancy and rotational DOFs removed.
Key Changes from Original 4+1 DOF Design:
Suppressed redundant parameter γ (Leg 5 replaced by 3).
Eliminated rotational DOF φ (double parallelogram linkages removed).

Simplified Jacobian Matrices

Relate joint velocities to Cartesian velocities:
$$
\mathbf{J} \dot{\mathbf{c}} = \mathbf{K} \dot{\boldsymbol{\theta}}
$$
- Variables:
- $\dot{\mathbf{c}} = [\dot{x}, \dot{y}, \dot{z}]^T$: End-effector translational velocities.
- $\dot{\boldsymbol{\theta}} = [\dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3]^T$: Joint angular velocities.

Singularities

Type I (Inverse Kinematics): Occurs when legs are fully extended/folded.
Type II (Forward Kinematics): Eliminated due to removed rotational constraints.

Inverse Kinematics

Joint Angle Expressions $( \theta_i )$

For legs $i = 1, 2, 3$:
$$
\theta_i = 2 \arctan\left( \frac{B_i \pm \sqrt{B_i^2 - (A_i + C_i)(C_i - A_i)}}{A_i + C_i} \right)
$$
Geometric Parameters:
For legs $i = 1, 2$: $$
\begin{align}
A_i &= 2(a_{iy} - b_{iy}) \
B_i &= 2(b_{ix} - a_{ix}) \
C_i &= \frac{\ell_{i2}^2 - \ell_{i1}^2 - (a_i - b_i)^T (a_i - b_i)}{\ell_{i1}}
\end{align}
$$
For leg $i = 3$: $$
\begin{align}
A_i &= 2(a_{iz} - b_{iz}) \
B_i &= 2(b_{ix} - a_{ix}) \
C_i &= \frac{\ell_{i2}^2 - \ell_{i1}^2 - (a_i - b_i)^T (a_i - b_i)}{\ell_{i1}}
\end{align}
$$
Variables:
- $a_i$ : Base joint position of leg $i$.
- $b_i$ : Distal joint position of leg $i$.
- $\ell_{i1}, \ell_{i2}$: Proximal/distal link lengths.

Workspace

Translational Workspace

Dimensions: ~0.5m × 1.0m × 0.3m (x, y, z).
Limits:
Determined by leg lengths ($\ell_{i1}, \ell_{i2}$).
Constrained by joint ranges (Fig. 10 in the paper).

Performance

Speed: Up to 3.5 m/s.

Velocity Equations

Angular Velocity $\dot{\theta}_i$

Derived from Jacobian matrices:
$$
\dot{\boldsymbol{\theta}} = \mathbf{K}^{-1} \mathbf{J} \dot{\mathbf{c}}
$$

Original Paper:
L. -T. Schreiber and C. Gosselin, "Schönflies Motion PARAllel Robot (SPARA): A Kinematically Redundant Parallel Robot With Unlimited Rotation Capabilities," in IEEE/ASME Transactions on Mechatronics, 2019.
DOI: 10.1109/TMECH.2019.2929646.

2. Dynamics

Abstract

This simulation uses the Equivalent Point Mass (EPM) method — a simplified approach for dynamic modeling of parallel robots. The method replaces distal links with dynamically equivalent point masses at their endpoints, eliminating complex angular velocity calculations. Derived from Lagrangian mechanics, this technique is computationally efficient while maintaining good accuracy for slender links.

Equivalent Point Mass (EPM) Method

Core Concept

The EPM method simplifies dynamics by:

Replacing distal links with point masses at their endpoints (Bᵢ and Cᵢ)
Assigning masses based on energy equivalence principles
Handling potential and kinetic energy separately

Mass Assignment Strategies

For Potential Energy (exact equivalence):

\[ m_{p1} = \frac{m l_2}{l_1 + l_2}, \quad m_{p2} = \frac{m l_1}{l_1 + l_2} \]

Where $ l_1, l_2 $ are distances from the center of mass to the endpoints.
For symmetric links:

\[ m_{p1} = m_{p2} = \frac{m}{2} \]

For Kinetic Energy (approximate equivalence):

Fixed Mass Method:

$ m_{k1} = m_{k2} = \frac{m}{2} $ (simple but less accurate)
or $ m_{k1} = \frac{m}{3}, \quad m_{k2} = \frac{2m}{3} $ (better for rotational inertia)

Variable Mass Method (higher accuracy):

\[ \begin{align*} m_{k1} &= \frac{m}{3}(1 + k_1) \\ m_{k2} &= \frac{m}{3}(1 + k_2) \end{align*} \]

With $ k $-coefficients calculated via minimum-norm solution:

\[ k_1 = \frac{|\mathbf{v}_1|^2}{|\mathbf{v}_1|^4 + |\mathbf{v}_2|^4} \mathbf{v}_1^T \mathbf{v}_2, \quad k_2 = \frac{|\mathbf{v}_2|^2}{|\mathbf{v}_1|^4 + |\mathbf{v}_2|^4} \mathbf{v}_1^T \mathbf{v}_2 \]

Dynamic Model Implementation

Energy Formulation

Kinetic energy for distal links simplifies to:

\[ T_{\text{EPM}} = \frac{1}{2} (m_{k1} \mathbf{v}_1^T \mathbf{v}_1 + m_{k2} \mathbf{v}_2^T \mathbf{v}_2) \]

Actuator Torques

Total torque combines contributions from:

Proximal links (joint space):

\[ \boldsymbol{\tau}_a = \frac{d}{dt}\left( \frac{\partial T}{\partial \dot{\boldsymbol{\theta}}} \right) - \frac{\partial T}{\partial \boldsymbol{\theta}} + \frac{\partial V}{\partial \boldsymbol{\theta}} \]

Platform + distal points (Cartesian space):

\[ \mathbf{f} = \frac{d}{dt}\left( \frac{\partial T}{\partial \dot{\mathbf{p}}} \right) - \frac{\partial T}{\partial \mathbf{p}} + \frac{\partial V}{\partial \mathbf{p}} \]

Total actuator torque:

\[ \boldsymbol{\tau} = \boldsymbol{\tau}_a + (\mathbf{J}^{-1}\mathbf{K})^T \mathbf{f} \]

Advantages & Limitations

✔️ Advantages

Eliminates complex angular velocity calculations
Reduces computational cost by 30–60% vs exact models
Maintains >95% accuracy for slender links
Fixed mass method enables real-time control
Variable mass improves accuracy for known trajectories

❌ Limitations

Accuracy decreases for thick/rotating links
Fixed masses: limited accuracy across configurations
Variable masses: requires known trajectories
Potential energy: exact only when COM lies on link axis

Performance Comparison

Method	Comp. Cost	Torque RMSE	Best For
Fixed Mass EPM	Lowest	5–6 Nmm	Real-time control
Variable Mass	Medium	2–3 Nmm	Known trajectories
Slender Link*	High	0.4 Nmm	General-purpose

* Reference method from original paper

Conclusion

The EPM method provides an effective balance between accuracy and computational efficiency. For real-time control, fixed masses ($ m/2 $ or $ m/3 + 2m/3 $) are recommended. For trajectory-based tasks, variable masses offer superior accuracy with moderate computation overhead.

Original Paper: Zhou Z., Gosselin C. (2024) Simplified Inverse Dynamic Models of Parallel Robots Based on Lagrangian Approach.
Meccanica, 59:657–680. DOI: 10.1007/s11012-024-01782-6