In the absence of work on the system, is there a closed form equation for the motion of a set of constrained rigid bodies (let's say, using Revolute (ie: simple pivot) constraints)?

If the bodies are fixed to the ground and subject to gravity, there definitely isn't. For instance, the double pendulum. But if no work is being done (ie: there are no non-constraint forces) does the motion allow a closed form? If so, how do you find it given the initial velocities and configuration of the system?

The center of mass of the system would have a linear velocity, for instance, as a consequence of Newton's second law. But the individual bodes themselves could have some very twisty motions.

I'm thinking it would have a form similar to the forward kinematics equation for a robot arm. That has the form $[\Theta_1] * (r_1 + [\Theta_2] * (r_2 + \ldots))$, where $r_i$ is the displacement from one link to the next and $[\Theta_i]$ is the rotation transform of a certain link.

EDIT: as an example, consider something like:

frisbee with pendulums

Are the motions of the central body and all the little pendulums expressible in a closed form assuming there's no gravity or motors or anything?

  • $\begingroup$ Yes it is called the recursive newton euler equations. Look it up. $\endgroup$ Jan 27, 2014 at 19:38
  • $\begingroup$ I can still only find links to numerical methods to solve the ODE for systems being driven by outside forces. $\endgroup$
    – Jay Lemmon
    Jan 27, 2014 at 19:51
  • $\begingroup$ Follow course 6 here. If the external forcing is zero, then the method still stands. It is based on the principle of least action. $\endgroup$ Jan 27, 2014 at 19:55
  • $\begingroup$ BTW, there is a closed form solution to the double pendulum. In fact there are closed form solutions to lots of problems, it depends on your apetitle on when complex is too complex. $\endgroup$ Jan 27, 2014 at 19:58
  • $\begingroup$ By closed form I mean I want to find the configuration of the system in the future in constant time. So no iterative algorithms where the number of iterations depends on how far in to the future I want to find the configuration for. $\endgroup$
    – Jay Lemmon
    Jan 27, 2014 at 20:05

1 Answer 1


The no work done requirement is translated at each prismatic joint as $$\vec{z} \cdot \vec{F} = 0$$ where $\vec{z}$ is the motion axis of the joint, and $\vec{F}$ the force vector through the joint. For a revolute joint the equation is $$\vec{z}\cdot \vec{\tau}=0$$ with $\vec{\tau}$ the torques through the joint.

If you have $N$ joints, these are $N$ equations to be solved for the $N$ joint accelerations $\ddot{q}$.

In general there is not analytic solution, only for special cases

NOTE: The rotational velocity vector resulting from a sequence of rotations $R_1 R_2 \ldots R_N$ is

$$\vec{\omega} = \dot{q}_1 \vec{z}_1 + R_1 \left( \dot{q}_2 \vec{z}_2 + R_2 \left( \ldots + R_{N-1} \dot{q}_N \vec{z}_N\right)\right) $$

Once the joint velocities $\dot{q}$ are known, the body velocities $\vec{\omega}$ are computed as above.


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