Duc-Tri This post is part of my Robotics Modeling, Control, and Design Tutorial series.
Introduction to Dynamics
While kinematics describes how a robot moves, dynamics explains why it moves that way. In other words:
- Kinematics focuses on position, velocity, and acceleration.
- Dynamics includes mass, inertia, and forces/torques, and describes how motion is generated in response to those quantities.
Dynamics is essential for simulating physical behavior, designing control laws, and analyzing stability.
General Form of Robot Dynamics
The dynamics of the robot are described by a system of nonlinear Ordinary Differential Equations (ODEs) as follows (assuming that there is no force at the end effector):
\[\begin{align} \mathbf{M}(\boldsymbol{\chi}) \ddot{\boldsymbol{\chi}} + \mathbf{C}(\boldsymbol{\chi}, \dot{\boldsymbol{\chi}}) \dot{\boldsymbol{\chi}} + \mathbf{g}(\boldsymbol{\chi}) = \boldsymbol{\tau} \end{align}\]where:
- \(\boldsymbol{\chi} \in \mathbb{R}^n\): Vector of generalized coordinates, typically joint angles for revolute joints.
- \(\dot{\boldsymbol{\chi}},\, \ddot{\boldsymbol{\chi}} \in \mathbb{R}^n\): First and second derivatives of joint positions, representing joint velocities and accelerations.
- \(\boldsymbol{\tau} \in \mathbb{R}^n\): Vector of input torques applied at the joints by actuators.
- \(\mathbf{M}(\boldsymbol{\chi}) \in \mathbb{R}^{n \times n}\): Inertia matrix, a symmetric and positive-definite matrix that encodes the mass and geometry of the robot.
- \(\mathbf{C}(\boldsymbol{\chi}, \dot{\boldsymbol{\chi}}) \dot{\boldsymbol{\chi}} \in \mathbb{R}^n\): Coriolis and centrifugal term, capturing the nonlinear velocity-dependent effects.
- \(\mathbf{g}(\boldsymbol{\chi}) \in \mathbb{R}^n\): Vector of generalized forces due to gravity, often omitted in planar cases where gravity acts orthogonally to the plane of motion.
Dynamic Model of 2R Plannar Robots
Lagrangian Formulation
To derive the dynamic equations of motion, we apply the Lagrangian mechanics method, which constructs the Lagrangian function:
\[\begin{align} \mathcal{L} = K - U \end{align}\]where:
- \(K\) is the total kinetic energy of the system
- \(U\) is the total potential energy due to gravity
The energy terms are computed by considering:
- Rotational kinetic energy due to link inertias \(I_1, I_2\)
- Translational kinetic energy from distributed link masses \(m_1^L, m_2^L\)
- Point masses located at joints and at the end-effector: \(m_1, m_2, m_E\)
- Gravitational potential energy of all mass components
1. Kinetic Energy
Let:
- \(\dot{\chi}_1, \dot{\chi}_2\) be the joint angular velocities
- \(\mathbf{v}_i\) be the linear velocity of the center of mass or a lumped point mass \(m_i\)
Then the total kinetic energy is given by:
\[\begin{align} K = \frac{1}{2} I_1 \dot{\chi}_1^2 + \frac{1}{2} I_2 (\dot{\chi}_1 + \dot{\chi}_2)^2 + \sum_i \frac{1}{2} m_i \| \mathbf{v}_i \|^2. \end{align}\]Here, \(\mathbf{v}_i\) is derived from forward kinematics (see Kinematics of Planar Robots with Revolute Joints). We have the planar positions of the second joint and the end-effector as:
\[\begin{align} x_2 &= l_1 \cos\chi_1,\\ y_2 &= l_1 \sin\chi_1, \\ x_E &= l_1 \cos\chi_1 + l_2 \cos(\chi_1 + \chi_2),\\ y_E &= l_1 \sin\chi_1 + l_2 \sin(\chi_1 + \chi_2). \end{align}\]Taking the time derivative of the positions gives the velocities:
Joint 2:
\[\begin{align} \dot{x}_2 &= -l_1 \sin\chi_1 \cdot \dot{\chi}_1 \\ \dot{y}_2 &= l_1 \cos\chi_1 \cdot \dot{\chi}_1. \end{align}\]Hence:
\[\begin{align} \| \mathbf{v}_2 \|^2 = \dot{x}_2^2 + \dot{y}_2^2 = l_1^2 \dot{\chi}_1^2. \end{align}\]End-Effector:
\[\begin{align} \dot{x}_E &= -l_1 \sin\chi_1 \cdot \dot{\chi}_1 -l_2 \sin(\chi_1 + \chi_2) \cdot (\dot{\chi}_1 + \dot{\chi}_2) \\ \dot{y}_E &= l_1 \cos\chi_1 \cdot \dot{\chi}_1 +l_2 \cos(\chi_1 + \chi_2) \cdot (\dot{\chi}_1 + \dot{\chi}_2). \end{align}\]Using the Pythagorean identity, we find:
\[\begin{align} \| \mathbf{v}_E \|^2 = l_1^2 \dot{\chi}_1^2 + l_2^2 (\dot{\chi}_1 + \dot{\chi}_2)^2 + 2 l_1 l_2 \cos(\chi_2) \dot{\chi}_1 (\dot{\chi}_1 + \dot{\chi}_2) \end{align}\]Substituting into the total expression for kinetic energy:
\[\begin{align} K &= \frac{1}{2} I_1 \dot{\chi}_1^2 + \frac{1}{2} I_2 (\dot{\chi}_1 + \dot{\chi}_2)^2 + \frac{1}{2} m_2 l_1^2 \dot{\chi}_1^2 \\ &\quad + \frac{1}{2} m_E \left[ l_1^2 \dot{\chi}_1^2 + l_2^2 (\dot{\chi}_1 + \dot{\chi}_2)^2 + 2 l_1 l_2 \cos\chi_2 \dot{\chi}_1 (\dot{\chi}_1 + \dot{\chi}_2) \right] \end{align}\]This provides the full analytical expression for the kinetic energy of the 2R planar manipulator, accounting for both distributed inertias and point masses.
2. Potential Energy
Assuming gravity acts in the negative \(y\) direction with magnitude \(g\), the gravitational potential energy is computed from the vertical positions of the masses.
Let:
- \(y_2\) be the vertical position of the second joint
- \(y_E\) be the vertical position of the end-effector
- \(m_2, m_E\) be the masses at joint 2 and the end-effector, respectively
Then the potential energy is:
\[\begin{align} U = m_2 g y_2 + m_E g y_E \end{align}\]From forward kinematics:
\[\begin{align} y_2 &= l_1 \sin(\chi_1), \\ y_E &= l_1 \sin(\chi_1) + l_2 \sin(\chi_1 + \chi_2). \end{align}\]Substituting into the expression:
\[\begin{align} U &= m_2 g \cdot l_1 \sin(\chi_1) + m_E g \left[ l_1 \sin(\chi_1) + l_2 \sin(\chi_1 + \chi_2) \right] \\ &= (m_2 + m_E) g l_1 \sin(\chi_1) + m_E g l_2 \sin(\chi_1 + \chi_2) \end{align}\]This expression captures the potential energy of the robot in terms of its configuration.
3. Euler–Lagrange Equations
For each generalized coordinate \(\chi_k\), the dynamics are governed by:
\[\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{\chi}_k} \right) - \frac{\partial \mathcal{L}}{\partial \chi_k} = \tau_k\]We now plug in the expressions for kinetic energy \(K\) and potential energy \(U\):
For Joint 1
The resulting equation is:
\[\begin{align} &\left[ I_1 + I_2 + (m_2 + m_E) l_1^2 + m_E l_2^2 + 2 m_E l_1 l_2 \cos(\chi_2) \right] \ddot{\chi}_1 \\ &+ \left[ I_2 + m_E l_2^2 + m_E l_1 l_2 \cos(\chi_2) \right] \ddot{\chi}_2 \\ &- m_E l_1 l_2 \sin(\chi_2) \left(2 \dot{\chi}_1 \dot{\chi}_2 + \dot{\chi}_2^2 \right) \\ &+ (m_2 + m_E) g l_1 \cos(\chi_1) + m_E g l_2 \cos(\chi_1 + \chi_2) \\ &= \tau_1 \end{align}\]For Joint 2
The resulting equation is:
\[\begin{align} &\left[ I_2 + m_E l_2^2 + m_E l_1 l_2 \cos(\chi_2) \right] \ddot{\chi}_1 \\ &+ \left[ I_2 + m_E l_2^2 \right] \ddot{\chi}_2 \\ &+ m_E l_1 l_2 \sin(\chi_2) \dot{\chi}_1^2 \\ &+ m_E g l_2 \cos(\chi_1 + \chi_2) \\ &= \tau_2 \end{align}\]These two coupled second-order nonlinear differential equations fully describe the dynamics of the 2R planar robot under the influence of gravity, link inertia, and external torques \(\tau_1, \tau_2\).
Summary
This post introduced the dynamics of a 2R planar robot using the Lagrangian method. We derived expressions for:
- Kinetic and potential energy based on joint angles, link lengths, and masses.
- Euler–Lagrange equations that lead to two nonlinear second-order differential equations.
- The compact MCG form:
This formulation provides the basis for simulation, control, and analysis of robot motion.
Kinematics of Planar Robots with Revolut Joints