Math Problem Statement

give me the mass matrix of Stanford arm robot which is 3 degree of freedom robot (RRP)

Solution

The Stanford arm robot, an RRP (Revolute-Revolute-Prismatic) robot, has a mass (inertia) matrix M(q)M(q) that represents the kinetic energy terms due to its joint velocities. For a 3-DOF RRP robot, the mass matrix is derived from the kinetic energy expression based on the robot's link parameters and joint configurations.

Notation:

  • q1q_1, q2q_2: Revolute joint angles.
  • q3q_3: Prismatic joint displacement.
  • mim_i: Mass of the ii-th link.
  • IziI_{zi}: Moment of inertia of the ii-th link about the z-axis (rotational axes for q1q_1, q2q_2).
  • d3d_3: Length of the prismatic extension (variable q3q_3).
  • riGr_{iG}: Distance from the ii-th link's axis to its center of mass.

General Form of the Mass Matrix:

The mass matrix M(q)M(q) for a 3-DOF robot arm can be expressed as:

\begin{bmatrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33} \end{bmatrix}$$ #### Elements of the Mass Matrix: 1. **Diagonal Terms**: - $$M_{11}$$: Inertia about $$q_1$$, includes $$I_{z1}$$, $$I_{z2}$$, and contributions from masses $$m_2$$, $$m_3$$. - $$M_{22}$$: Inertia about $$q_2$$, includes $$I_{z2}$$ and $$m_3$$ terms influenced by $$q_3$$. - $$M_{33}$$: Translational kinetic energy term due to the prismatic joint, directly proportional to $$m_3$$. 2. **Off-Diagonal Terms**: - $$M_{12}$$, $$M_{21}$$: Coupling between $$q_1$$ and $$q_2$$. - $$M_{13}$$, $$M_{31}$$: Coupling between $$q_1$$ and $$q_3$$. - $$M_{23}$$, $$M_{32}$$: Coupling between $$q_2$$ and $$q_3$$. #### Simplified Expression: For the Stanford arm, considering the link parameters and ignoring Coriolis and centrifugal terms for clarity: $$M(q) = \begin{bmatrix} I_{z1} + I_{z2} + m_2 r_{2G}^2 + m_3 (d_3^2 + q_3^2) & I_{z2} + m_3 d_3 q_3 & m_3 d_3 \\ I_{z2} + m_3 d_3 q_3 & I_{z2} + m_3 (d_3^2 + q_3^2) & m_3 q_3 \\ m_3 d_3 & m_3 q_3 & m_3 \end{bmatrix}$$ This matrix captures the dynamics of the Stanford arm's three degrees of freedom. Would you like me to elaborate on deriving this matrix or explain any particular term? --- ### 5 Related Questions: 1. How does the mass matrix change with variations in the prismatic joint displacement ($$q_3$$)? 2. What role do the Coriolis and centrifugal terms play in the dynamic equations of motion? 3. How are the kinetic and potential energy terms used to derive the Lagrangian for a robotic manipulator? 4. Can the coupling terms in $$M(q)$$ create difficulties in controlling the robot? 5. How does the inclusion of gravity affect the dynamics of the Stanford arm? ### Tip: To analyze and simulate the dynamics of robotic systems, software tools like MATLAB or Python's SymPy library are invaluable for deriving and validating the mass matrix.

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Math Problem Analysis

Mathematical Concepts

Robotic Dynamics
Mass Matrix
Kinetic Energy
Lagrangian Dynamics

Formulas

M(q) = [[M11, M12, M13], [M21, M22, M23], [M31, M32, M33]]
M11 = I_{z1} + I_{z2} + m2 * r_{2G}^2 + m3 * (d3^2 + q3^2)
M22 = I_{z2} + m3 * (d3^2 + q3^2)
M33 = m3
Off-diagonal terms: M12 = M21, M13 = M31, M23 = M32

Theorems

Lagrange's Equations of Motion
Kinetic Energy and Inertia Matrix Relationship

Suitable Grade Level

Undergraduate Robotics or Mechanical Engineering