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Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments


Coordinate singularities and gimbal lock are two phenomena that present themselves in models for the dynamics of mechanical systems. The former phenomenon pertains to the coordinates used to parameterize the configuration manifold of the system, while the latter phenomenon has a distinctive physical manifestation. In the present paper, we use tools from differential geometry to show how gimbal lock is intimately associated with an orthogonality condition on the applied forces and moments which act on the system. This condition is equivalent to a generalized applied force being normal to the configuration manifold of the system. Numerous examples, including the classic bead on a rotating hoop example and a gimbaled rigid body, are used to illuminate the orthogonality condition. These examples help to offer a new explanation for the elimination of gimbal lock by the addition of gimbals and demonstrate how integrable constraints alter the configuration manifold and may consequently eliminate coordinate singularities.

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  1. For a comprehensive reference on gyroscopic devices mechanically suspended in gimbals, see [1]. For a technical overview of the inertial platforms used in the Apollo spacecraft missions, see [19] and [29]. Other useful references on inertial navigation include [8], [21], and [30, Chap. 6].

  2. Additional background for a variety of sources on rotation tensors, Euler angles, Euler bases, and dual Euler bases are collected in Appendix.

  3. In other words, the generalized constraint forces do no virtual work in any motion of the system compatible with the constraints.

  4. For representations of rotations with constant angular velocities on the real projective 2-space \(\mathbb{R}P^{2}\), the reader is referred to [20].

  5. A discussion of the constraint forces and moments acting on the bodies shown in this figure can be found in [24].


  1. Arnold, R.N., Maunder, L.: Gyrodynamics and Its Engineering Applications. Academic Press, New York (1961)

    MATH  Google Scholar 

  2. Baruh, H.: Analytical Dynamics. WCB/McGraw–Hill, Boston (1999)

    Google Scholar 

  3. Bohigas, O., Manubens, M., Ros, L.: Singularity-free path planning. In: Singularities of Robot Mechanisms, pp. 111–136. Springer, Berlin (2017)

    Chapter  Google Scholar 

  4. Casey, J.: Geometrical derivation of Lagrange’s equations for a system of particles. Am. J. Phys. 62(9), 836–847 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  5. Casey, J.: On the advantages of a geometrical viewpoint in the derivation of Lagrange’s equations for a rigid continuum. In: Casey, J., Crochet, M.J. (eds.) Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids: A Collection of Papers in Honor of Paul M. Naghdi, pp. 805–847. Birkhäuser, Basel (1995).

    Chapter  Google Scholar 

  6. Casey, J.: On the representation of rigid body rotational dynamics in Hertzian configuration space. Int. J. Eng. Sci. 49(12), 1388–1396 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  7. Casey, J., O’Reilly, O.M.: Geometrical derivation of Lagrange’s equations for a system of rigid bodies. Math. Mech. Solids 11(4), 401–422 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  8. Fernandez, M., Macomber, G.R.: Inertial Guidance Engineering. Prentice Hall, Englewood Cliffs (1962)

    Google Scholar 

  9. Ganovelli, F., Corsini, M., Pattanaik, S., Di Benedetto, M.: Introduction to Computer Graphics: A Practical Learning Approach. CRC Press, Boca Raton (2015)

    MATH  Google Scholar 

  10. Greenwood, D.T.: Principles of Dynamics. Prentice Hall, Englewood Cliffs (1988)

    Google Scholar 

  11. Greenwood, D.T.: Advanced Dynamics. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  12. Guha, S.: Computer Graphics Through OpenGL: From Theory to Experiments. CRC Press, Boca Raton (2015)

    MATH  Google Scholar 

  13. Hanson, A.J.: Visualizing quaternions. In: ACM SIGGRAPH 2005 Courses, p. 1. ACM, New York (2005)

    Google Scholar 

  14. Kane, T.R., Likins, P.W., Levinson, D.A.: Spacecraft Dynamics. McGraw–Hill, New York (1983), 445 p.

    Book  Google Scholar 

  15. Machover, C.: Basics of Gyroscopes, vol. 1. JF Rider, New York (1960)

    Google Scholar 

  16. Marsden, J.E., Ratiu, T.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17. Springer, Berlin (2013)

    MATH  Google Scholar 

  17. McConnell, A.J.: Applications of the Absolute Differential Calculus. Blackie, London (1947). Corrected reprinted edition

    Google Scholar 

  18. Mebius, J.E., Kasperink, H.R., Kooijman, M.D.: Mathematics for simulation of near-vertical aircraft attitudes. In: Making It Real: Proceedings CEAS Symposium on Simulation Technology, Confederation of European Aerospace Societies, Delft, The Netherlands pp. 1–11 (1995). Paper Number MOD04

    Google Scholar 

  19. Moore, R.L., Thomason, H.E.: Gimbal geometry and attitude sensing of the ST-124 stabilized platform. Technical Report No. 19620002325 (1962)

  20. Novelia, A., O’Reilly, O.M.: On geodesics of the rotation group SO(3). Regul. Chaotic Dyn. 20(6), 729–738 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  21. O’Donnell, C.F.: Inertial Navigation: Analysis and Design. McGraw–Hill, New York (1964)

    Google Scholar 

  22. O’Reilly, O.M.: The dual Euler basis: constraints, potentials, and Lagrange’s equations in rigid body dynamics. ASME J. Appl. Mech. 74(2), 256–258 (2007).

    Article  MATH  Google Scholar 

  23. O’Reilly, O.M.: Intermediate Dynamics for Engineers: A Unified Treatment of Newton–Euler and Lagrangian Mechanics. Cambridge University Press, New York (2008)

    Book  MATH  Google Scholar 

  24. O’Reilly, O.M., Srinivasa, A.R.: A simple treatment of constraint forces and constraint moments in the dynamics of rigid bodies. ASME Appl. Mech. Rev. 67(1), 014,801 (2014).

    Article  Google Scholar 

  25. Shoemake, K.: Animating rotation with quaternion curves. ACM SIGGRAPH Comput. Graph. 19(3), 245–254 (1985)

    Article  Google Scholar 

  26. Shuster, M.D., Oh, S.: Three-axis attitude determination from vector observations. J. Guid. Control Dyn. (2012)

  27. Stuelpnagel, J.: On the parametrization of the three-dimensional rotation group. SIAM Rev. 6(4), 422–430 (1964)

    MathSciNet  Article  MATH  Google Scholar 

  28. Synge, J.L., Schild, A.: Tensor Calculus. University of Toronto Press, Toronto (1949)

    MATH  Google Scholar 

  29. Thomason, H.E.: A general description of the ST124-M intertial platform system. Tech. rep., George C. Marshall Space Flight Center, Huntsville, Alabama, Washington, DC (1965). Technical Report No. 19650024833

  30. Thomson, W.T.: Introduction to Space Dynamics. Dover, New York (1986)

    Google Scholar 

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The work of Evan Hemingway was supported by a Berkeley Fellowship from the University of California at Berkeley and a U.S. National Science Foundation Graduate Research Fellowship. The authors are grateful to Professor James Casey for his helpful comments on an earlier draft of this paper.

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Correspondence to Oliver M. O’Reilly.

Appendix: Background on rotations

Appendix: Background on rotations

An example of a 3-dimensional configuration manifold results from the constrained motion of a rigid body rotating about a fixed point, such as in the motion of the Lagrange top. In this case, the rotation tensor \(\mathbf{Q}\) of the top belongs to the proper-orthogonal subgroup of linear second-order tensors, a 3-dimensional manifold called \(\mathsf{Orth}^{+}\) embedded in \(\mathcal{C}^{9}\). This subgroup is isomorphic to \(\mathit{SO}(3)\), the special orthogonal group of \(3 \times 3\) matrices. Furthermore, \(\mathit{SO}(3)\) is diffeomorphic to \(\mathbb{R}P^{3}\), real projective 3-space, and we can associate the configuration of the top to a point in \(\mathbb{R}P^{3}\).Footnote 4

Given an angle \(\theta \) and axis of rotation \(\mathbf{p}\), Euler’s formula for the rotation tensor is

$$ {\mathbf{Q}} = \mathbf{L} ( \theta , \mathbf{p} ) = \cos ( \theta ) ( \mathbf{I} - \mathbf{p} \otimes {\mathbf{p}} ) + \sin ( \theta ) \text{skwt} ( \mathbf{p} ) + \mathbf{p} \otimes {\mathbf{p}}, $$

where \(\text{skwt} ( \mathbf{p} ) \) indicates the skew tensor of \(\mathbf{p}\), which is given by the mapping

$$ \text{skwt} ( \mathbf{p} ) = - \boldsymbol {\epsilon }{\mathbf{p}}, $$

where \(\boldsymbol {\epsilon }\) is the third-order alternator tensor. Since \(\mathbf{Q}\) is proper orthogonal, it satisfies \(\mathbf{Q} {\mathbf{Q}} ^{T} = \mathbf{I}\) and \(\det {\mathbf{Q}} = + 1\). A skew-symmetric angular velocity tensor may therefore be defined using \(\mathbf{Q}\) as \(\boldsymbol {\Omega }= \dot{\mathbf{Q}} {\mathbf{Q}}^{T}\). We denote the space of skew-symmetric tensors as \(\mathsf{Skw}\). The angular velocity vector \(\boldsymbol {\upomega }\) is the axial vector of \(\boldsymbol {\Omega }\) which is given by the mapping

$$ \boldsymbol {\upomega }= \text{ax} ( \boldsymbol {\Omega }) = -\frac{1}{2} \boldsymbol {\epsilon }[ \boldsymbol {\Omega }] . $$

Thus, there exists an isomorphism between \(\mathsf{Skw}\) and \(\mathbb{E}^{3}\) through the maps \(\text{ax} ( \cdot ) \) and \(\text{skwt} ( \cdot ) \) for which the following is true:

$$ \boldsymbol {\Omega }{\mathbf{a}} = \boldsymbol {\upomega }\times \mathbf{a} $$

for any \(\mathbf{a} \in \mathbb{E}^{3}\). Euler angles are a common choice of parametrization for \(\mathbf{Q}\). A general Euler angle decomposition is given by

$$ {\mathbf{Q}} = \mathbf{L} \bigl( \nu ^{3}, \mathbf{g}_{3} \bigr) \mathbf{L} \bigl( \nu ^{2}, \mathbf{g}_{2} \bigr) \mathbf{L} \bigl( \nu ^{1}, \mathbf{g}_{1} \bigr) , $$

where \(\{ \nu ^{1}, \nu ^{2}, \nu ^{3} \} \) are the Euler angles and \(\{ \mathbf{g}_{1}, \mathbf{g}_{2}, \mathbf{g}_{3} \} \) are the corresponding axes of rotation, which are collectively known as the Euler basis [2224]. It is also convenient to define a reciprocal basis \(\{ \mathbf{g}^{i} \} \) known as the dual Euler basis with the property

$$ {\mathbf{g}}_{i} \cdot {\mathbf{g}}^{j} = \delta _{i}^{j} \quad ( i,j =1, 2, 3 ) , $$

where \(\delta _{i}^{j}\) is the Kronecker delta. A corotational (or body-fixed) basis is defined as the set \(\{ \mathbf{e}_{i} \} \) for which \(\mathbf{e}_{i} = \mathbf{Q} {\mathbf{E}}_{i}\) for \(i =1,2,3\). The rotation tensor has the same components in the corotational basis as it does in the inertial basis: \(\mathbf{Q} = Q _{ij} {\mathbf{e}}_{i} \otimes {\mathbf{e}}_{j} = Q_{ij} {\mathbf{E}} _{i} \otimes {\mathbf{E}}_{j}\). Using the abbreviations \(\cos ( \nu ) = \text{c} \nu \) and \(\sin ( \nu ) = \text{s} \nu \), a 3–2–1 Euler angle set yields the following parametrization:

$$ [ Q_{ij} ] = \left [ \textstyle\begin{array}{c @{\quad}c@{\quad} c} \text{c}\nu ^{1} \text{c}\nu ^{2} & \text{c}\nu ^{1} \text{s} \nu ^{2} \text{s} \nu ^{3} - \text{s} \nu ^{1} \text{c}\nu ^{3} & \text{c}\nu ^{1} \text{s} \nu ^{2} \text{c}\nu ^{3} + \text{s} \nu ^{1} \text{s} \nu ^{3} \\ \text{s} \nu ^{1} \text{c} \nu ^{2} & \text{s} \nu ^{1} \text{s} \nu ^{2} \text{s} \nu ^{3} + \text{c} \nu ^{1} \text{c} \nu ^{3} & \text{s} \nu ^{1} \text{s} \nu ^{2} \text{c} \nu ^{3} - \text{c} \nu ^{1} \text{s} \nu ^{3} \\ -\text{s} \nu ^{2} & \text{c} \nu ^{2} \text{s} \nu ^{3} & \text{c} \nu ^{2} \text{c} \nu ^{3} \end{array}\displaystyle \right ] , $$

where \(\nu ^{1} \in [0, 2\pi )\), \(\nu ^{2} \in [ -\frac{\pi }{2}, \frac{\pi }{2} ] \), and \(\nu ^{3} \in [0, 2\pi )\). A 3–1–3 Euler angle set yields the following parametrization:

$$ [ Q_{ij} ] = \begin{bmatrix} \text{c} \nu ^{1} \text{c} \nu ^{3} - \text{s} \nu ^{1} \text{c} \nu ^{2} \text{s} \nu ^{3} & - \text{c} \nu ^{1} \text{s} \nu ^{3} - \text{s} \nu ^{1} \text{c} \nu ^{2} \text{c} \nu ^{3} & \text{s} \nu ^{1} \text{s} \nu ^{2} \\ \text{s} \nu ^{1} \text{c} \nu ^{3} + \text{c} \nu ^{1} \text{c} \nu ^{2} \text{s} \nu ^{3} & - \text{s} \nu ^{1} \text{s} \nu ^{3} + \text{c} \nu ^{1} \text{c} \nu ^{2} \text{c} \nu ^{3} & - \text{c} \nu ^{1} \text{s} \nu ^{2} \\ \text{s} \nu ^{2} \text{s} \nu ^{3} & \text{s} \nu ^{2} \text{c} \nu ^{3} & \text{c} \nu ^{2} \end{bmatrix} , $$

where \(\nu ^{1} \in [0,2\pi )\), \(\nu ^{2} \in [ 0, \pi ] \), and \(\nu ^{3} \in [0, 2\pi )\).

For a given Euler angle parametrization, the following identity may be shown to hold:

$$ \frac{\partial {\mathbf{Q}}}{\partial \nu ^{i}} {\mathbf{Q}}^{T} = \text{skwt} ( \mathbf{g}_{i} ) \quad ( i =1,2,3 ) , $$

where \(\mathbf{Q}\) is given by (A.5).

A.1 Constraining one of the Euler angles

An interesting situation occurs when the first or third Euler angle is constrained and the configuration manifold for the rotational motion of the rigid body becomes a two-torus \(T^{2}\) provided \(\nu ^{2}\) is extended to \(\nu _{e}^{2} \in [0, 2\pi )\). The latter case arises in the motion of the rigid bodies shown in Fig. 2.Footnote 5 If a set of 3–1–3 Euler angles are used to parameterize the rotation tensor \(\mathbf{Q}\) of these bodies, then the third angle \(\nu ^{3} = 0\). To use two angles to parameterize the rotation \(\mathbf{Q}\) of the constrained rigid bodies shown in Fig. 2, it is necessary that

$$ {\mathbf{Q}} = \mathbf{L} \bigl( \nu ^{2}_{e}, \mathbf{e}_{1} \bigr) \mathbf{L} \bigl( \nu ^{1}, \mathbf{E}_{3} \bigr) , $$

In contrast to the unconstrained case (A.5), it is necessary to extend the range of the second angle: \(\nu ^{2} \to \nu ^{2}_{e} \in [0, 2\pi )\). The necessity of this extension can also be demonstrated by taking a pen and placing it on a horizontal surface. Fixing the orientation of the longitudinal axis of the pen is equivalent to fixing \(\nu ^{1}\). Then, by rotating the pen about its longitudinal axis, the necessity of having the extended angle \(\nu ^{2}_{e}\) can be readily seen.

A.2 A fixed axis of rotation

Suppose \(\mathbf{Q}\) is constrained so that the rotation axis is fixed, \(\mathbf{p} = \mathbf{p}_{0}\). Then, the configuration manifold is a 1-dimensional circle \(S^{1}\) contained in \(\mathsf{Orth}^{+}\): \(\mathbf{p}_{0}\otimes {\mathbf{p}}_{0}\) is normal to the plane of the circle and \(\cos ( \theta ) ( \mathbf{I} - \mathbf{p}_{0}\otimes {\mathbf{p}}_{0} ) + \sin ( \theta ) \text{skwt} ( \mathbf{p}_{0} ) \) is a radius vector to a point on the circle. A tensor spanning the tangent space to \(S^{1}\) is

$$ {\mathbf{a}}_{1} = \frac{\partial {\mathbf{Q}}}{\partial \theta } = - \sin ( \theta ) ( \mathbf{I} - \mathbf{p}_{0} \otimes {\mathbf{p}}_{0} ) + \cos ( \theta ) \, \text{skwt} ( \mathbf{p}_{0} ) , $$

which is clearly tangent to the circle.

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Hemingway, E.G., O’Reilly, O.M. Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments. Multibody Syst Dyn 44, 31–56 (2018).

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  • Gimbal lock
  • Coordinate singularities
  • Configuration manifold
  • Differential geometry
  • Analytical mechanics