[Modern Robotics] 강좌 3: 로봇 동역학 #1

 

 

 

 

+)This chapter deals with open chains, n-joint serial robots.

 

 

 

+)Math fundamentals

Positive-definite: A positive matrix can be regarded as a positive value. (My own explanation, uncertain)

For $A\in {\mathbb{R}}^{n\times n},x\in {\mathbb{R}}^{\mathbb{n}},x\ne 0$, A is a positive-definite matrix if $x^{T}Ax>0$

●If A is a positive-definite matrix, $det\left[\begin{array}{c}{a}_{11}\end{array}\right]>0$, $det\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]>0,\cdots ,det\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ & ⋮& \\ a_{n1}& \cdots & a_{nn}\end{array}\right]=det(A)>0$ 

●All eigenvalues are positive.

●e.g. Mass matrix, Stiffness matrix, Inertia matrix, etc.

 

 

Christoffel sybols of the first kind

${\mathrm{\Gamma }}_{ijk}(\theta )=\frac{1}{2}(\frac{\partial m_{ij}}{\partial {\theta }_k}+\frac{\partial m_{ik}}{\partial {\theta }_j}-\frac{\partial m_{jk}}{\partial {\theta }_i})$

●For inverse dynamics equations, $m$ indicates a element of mass matrix.

●$\mathrm{\Gamma }(\theta )\in {\mathbb{R}}^{n\times n\times n}$ consist of ${\mathrm{\Gamma }}_{ijk}(\theta )$ ($1\le i,j,k\le n$)

●${\mathrm{\Gamma }}_i(\theta )\in {\mathbb{R}}^{n\times n}$ consist of ${\mathrm{\Gamma }}_{ijk}(\theta )$ ($1\le i,j,k\le n$)

 

 

 

 

 

 

 

Inverse dynamics: $\text{Given }\theta ,\dot{\theta }\text{ and }\ddot{\theta }\in {\mathbb{R}}^n,\text{ find }\tau \in {\mathbb{R}}^n$

●$\tau =M(\theta )\ddot{\theta }+h(\theta ,\dot{\theta })$

 

Forward dynamics: $\text{Given }\theta ,\dot{\theta }\text{ and }\tau \in {\mathbb{R}}^n,\text{ find }\ddot{\theta }\in {\mathbb{R}}^n$

●$\ddot{\theta }=M^{-1}(\theta )(\tau -h(\theta ,\dot{\theta }))$

 

Lagrangian dynamics formulation

Lagrangian mechanics: Analyze motions using the energy.

$\mathcal{L}(q,\dot{q})=\sum _{i=1}^n{\mathcal{L}}_i=\sum _{i=1}^n({\mathcal{K}}_i(q,\dot{q})-{\mathcal{P}}_i(q))$

●Lagrangian $\mathcal{L}$ is the system's energy term (Kinetic energy - Potential energy). ($q$: Generalized coordinates)

 

 

Euler-Lagrangian equations with external forces. ($f$: Generalized forces)

$f_i=\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial {\dot{q}}_i}\right)-\frac{\partial \mathcal{L}}{\partial q_i}$

●The result of EoM must be the same as that of Newton's second law. (& F=m\frac{d v}{dt} &)

●Suitable for systems with fewer than 3-DoF.

 

 

Inverse dynamics with Lagrangian method: Derive inverse dynamics equation with Euler-Lagrange equation.

 

(Single joint): ${\tau }_i=\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial {\dot{\theta }}_i}\right)-\frac{\partial \mathcal{L}}{\partial {\theta }_i}=\sum _{j=1}^n{m}_{ij}(\theta )\ddot{\theta }+\sum _{j=1}^n\sum _{k=1}^n{\mathrm{\Gamma }}_{ijk}{\dot{\theta }}_j{\dot{\theta }}_k+\frac{\partial \mathcal{P}}{\partial {\theta }_i}$

 

(Entire joint): $\tau =M(\theta )\ddot{\theta }+c(\theta ,\dot{\theta })+g(\theta )=M(\theta )\ddot{\theta }+C(\theta ,\dot{\theta })\dot{\theta }+g(\theta )$       (${\tau }_i\in \mathbb{R}$,    $\tau ,\theta ,\dot{\theta },\ddot{\theta }\in {\mathbb{R}}^n$)

 

●If there are additional $\tau$ from end effector static wrench ${\mathcal{F}}_{tip}$,   $\tau =M(\theta )\ddot{\theta }+c(\theta ,\dot{\theta })+g(\theta )+J^T(\theta ){\mathcal{F}}_{tip}$

●For mass $M(\theta )$ and Coriolis matrix $C(\theta ,\dot{\theta })$, $\dot{M}-2C$ should be a skew-symmetric matrix.(Passivity property)

 

 

+)Explanation of mappings from acceleration to force below are only for the zero velocity case. (No Coriolis and centripetal effect)

Mass matrix $M(\theta )$: Represents the amount of (linear+rotational) inertia around each joint that changes depending on the joint configuration $\theta$.

●$M\in {\mathbb{R}}^{n\times n}$ contains the entire robot's both linear and rotational inertia.

●$m_{ij}$ represents (i,j)th element of Mass matrix $M(\theta )$

●Symmetric, Positive-definite

●Not simply represents mass and mass moment of inertia. (was my misconception)

●Mapping: $\ddot{\theta }⟶\tau$

●Unlike a point mass ($\tau =I\ddot{\theta },F=m\ddot{x}$), ${\tau }_i$ is generally not proportional to ${\ddot{\theta }}_i$ due to off-diagonal elements of $M(\theta )$.

 

End effector mass matrix $\mathrm{\Lambda }$: Represent the whole robot's effective inertia at the end effector. (Not good enough)

$\mathrm{\Lambda }(\theta )=J^T(\theta )M(\theta )J^{-1}(\theta )$

●Calculated from the mass matrix M

●Mapping: $\ddot{x}⟶f_x$,   ($x$: End effector configuration,   $f_x$: Force along the direction $x$)

 

 

 

 

Coriolis and Centripetal term $c(\theta ,\dot{\theta })$: Represent torques arise from the Coriolis and centripetal accelerations. These terms can occur even when $\ddot{\theta }=0$

 

 

Centripetal term: The centripetal force is essential for the mass to move in a circular motion and this makes torque on joints.

●Terms containing ${\dot{\theta }}_i^2$

●${\dot{\theta }}_i^2$ doesn't make any torque on joint $i$ because the centripetal's force direction.(Heading to center of joint $i$)

●Each joint have n-1 different centripetal terms. (${\dot{\theta }}_1^2{\dot{\theta }}_n^2$, except ${\dot{\theta }}_i^2$)

 

Coriolis term: The Coriolis force is applied to the mass when the mass is in radius-changing circular motion. (Not good enough)

●Terms containing ${\dot{\theta }}_i{\dot{\theta }}_j$

●${\dot{\theta }}_1\cdots {\dot{\theta }}_{i-1}$ don't make any torque on joint $i$.

●e.g. For n-joint open chain robots, if all joints are actuated, there will be ${}_n{\mathrm{C}}_2$ Coriolis terms on joint 1. (Uncertain)

 

Christoffel symbol: Represents the Coriolis and centripetal term on the joint i. (${\tau }_i=$~~~ equation)

●Index i: joint i

●${\mathrm{\Gamma }}_{ijk}$, $j=k$: Generate the centripetal term with ${\dot{\theta }}_j^2$ 

●${\mathrm{\Gamma }}_{ijk}$, $j\ne k$: Generate the Coriolis term with ${\dot{\theta }}_j{\dot{\theta }}_k$ 

●If mass matrix $M$ is independent of $\theta$, ${\mathrm{\Gamma }}_{ijk}(\theta )=0$ which means no Coriolis and centripetal force at all.

 

Coriolis matrix $C(\theta ,\dot{\theta })$: The $n\times n$ matrix for expressing Coriolis and Centripetal term $c(\theta ,\dot{\theta })$ as $C(\theta ,\dot{\theta })\dot{\theta }$ form.

An element of $C(\theta ,\dot{\theta })$ is $c_{ij}(\theta ,\dot{\theta })=\sum _{k=1}^n{\mathrm{\Gamma }}_{ijk}(\theta )\dot{{\theta }_k}$

 

 

 

 

Gravitational torque term $g(\theta )$: Represent torques arise from the gravity force.

$g(\theta )=\frac{\partial \mathcal{P}}{\partial \theta }$

●Holds only when gravity is the only conservative force applied to the open chain system. (e.g. Spring force that makes torque on joints)

 

 

 

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