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Articulated Rigid-Body Systems

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This chapter extends the techniques for rigid bodies to include articulated rigid bodies. Here, we shall focus on linking rigid bodies with joints that constraint their motion. Position-based, velocity-based and force-based constraints systems are explained and a framework is presented to derive the appropriate constraint equations based on the number of degree of freedoms associated with a joint. This framework is applied to spherical, universal, revolute, cylindrical, prismatic and rigid joints. However, these techniques can be easily applied to include other types of joints suitable to the reader’s interests.


  • Rigid Body
  • Contact Force
  • Anchor Point
  • Constraint Function
  • Revolute Joint

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  • DOI: 10.1007/978-1-4471-4417-5_5
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  1. 1.

    By convention, a positive joint force \(+\vec{F}_{i}\) is applied to body (B 1) i , whereas a negative joint force \(-\vec {F}_{i}\) is applied to body (B 2) i .

  2. 2.

    See Sect. 6.7 of Appendix A (Chap. 6) for more details on these expressions. and the auxiliary variable

  3. 3.

    The following analysis is still valid for any type of joint used.

  4. 4.

    Refer to Sect. 4.11.1 for a detailed discussion of the techniques used to compute the critical coefficient of friction.

  5. 5.

    By kinematical control we mean that the linear and angular position, velocity and acceleration of the bodies are obtained from an animation system, possibly by interpolating their values between two consecutive animation frames.


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Coutinho, M.G. (2013). Articulated Rigid-Body Systems. In: Guide to Dynamic Simulations of Rigid Bodies and Particle Systems. Simulation Foundations, Methods and Applications. Springer, London.

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