Consider a multibody robotic system with Lagrangian $$ L =\frac{1}{2} \dot{q}^{T}M\dot{q} - V $$ and equations of motion of the form $$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+A^{T}(q)\lambda+N(q)=0 $$

where $N=\frac{\partial V(q)}{\partial q}$ and $\lambda$ represents the Lagrange multipliers for the constraint forces (MLS pg. 269; pg. 287 in pdf). $A$ is the constraint matrix from the Pfaffian form $A\dot{q}=0$. We are concerned with holonomic constraints, which can be differentiated to obtain the Pfaffian form. $ C=\sum_{k=1}^{n} \Gamma_{ijk} \dot{q_{k}} $ where $\Gamma_{ijk}$ are the Christoffel symbols of the first kind.

The Lagrange multipliers can be calculated using the formula (MLS pg. 270, pg. 288 in pdf) $$\lambda =-(AM^{-1}A^{T})^{-1}AM^{-1}(C\dot{q}-N)$$

This gives the holonomic constraint forces $A^{T}\lambda$.

I would like to see some examples of this method for calculating the holonomic constraint forces for planar systems.

I have seen this method being applied to calculate the tension in a simple pendulum (MLS pg. 270, pg. 288 in pdf), but I would like to see examples of planar systems with more degrees of freedom.

I'm specifically interested in the case of a 2R manipulator shown below.

Planar 2R manipulator

The equations of motion are on pg. 164-165 (pg. 182-183 in the pdf) in MLS.

My objectives are:

  1. Apply the holonomic constraint $\theta_{1}(t)=0$ and obtain the holonomic constraint force.
  2. Show that we get the dynamics of a simple pendulum for the constrained system for initial conditions $\theta_{2}(0)=0, \dot{\theta_{2}}(0)=0$.

I have done the calculation for the constraint $\theta_{1}(t)=0$ but I'm not able to show 2. The expression I get for $\lambda$ is

$$\lambda= \begin{array}{c} \frac{g (d \text{m2} \text{r2} (\sin \text{t2})-\text{m2} \text{r2} (\cos \text{t2}) ((\cos \text{t2}) b+d))+b d (\sin \text{t2}) \dot{\text{t2}} \dot{\text{t2}}}{d} \\ \end{array} $$

where $(t1, t2)=(\theta_{1},\theta_{2})$ and $a, b, d$ are the geometric and inertial parameters $\alpha, \beta, \delta$ described in MLS (pg. 165; pg. 183 in the pdf).

Suggestions for other methods to calculate holonomic constraint forces would be helpful too.


MLS refers to

Murray, R. M., Li, Z., Sastry, S. S., & Sastry, S. S. (1994). A mathematical introduction to robotic manipulation. CRC press.


1 Answer 1


With multiple bodies it starts getting so complicated that I would forego doing it by hand and instead implement it in MATLAB or python using matrices. There are a few ways to do that (Lagrange isn't actually the most optimal one, as you're doing some unnecessary conversion to the energy domain), but it can be used. I could type out some equations, or post screenshots of my book. But my prof probably dos a far better job of explaining it:



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.