Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra \(\mathit{so}(3)\) of the rotation group \(\mathit{SO}(3)\). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on \(\mathit{so}(3)\) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.     

Forward Dynamics of Open-Loop Multibody Mechanisms Using an Efficient Recursive Algorithm Based on Canonical Momenta

A new method for establishing the equations of motion of multibodymechanisms based on canonical momenta is introduced in this paper.In absence of constraints, the proposed forward dynamicsformulation results in a Hamiltonian set of 2n first order ODEsin the generalized coordinates q and the canonical momenta p.These Hamiltonian equations are derived from a recursiveNewton–Euler formulation. As an example, it is shown how, in thecase of a serial structure with rotational joints, an O(n)formulation is obtained. The amount of arithmetical operations isconsiderably less than acceleration based O(n) formulations.     

Null Space Integration Method for Constrained Multibody Systems with No Constraint Violation

A method for integrating equations of motion of constrained multibodysystems with no constraint violation is presented. A mathematical model,shaped as a differential-algebraic system of index 1, is transformedinto a system of ordinary differential equations using the null-spaceprojection method. Equations of motion are set in a non-minimal form.During integration, violations of constraints are corrected by solvingconstraint equations at the position and velocity level, utilising themetric of the system's configuration space, and projective criterion to thecoordinate partitioning method. The method is applied to dynamicsimulation of 3D constrained biomechanical system. The simulation resultsare evaluated by comparing them to the values of characteristicparameters obtained by kinematic analysis of analyzed motion based onmeasured kinematic data.     

Structure of optimized generalized coordinates partitioned vectors for holonomic and non-holonomic systems

The generalized coordinates partitioning is a well-known procedure that can be applied in the framework of a numerical integration of the DAE systems. However, although the procedure proves to be a very useful tool, it is known that an optimization algorithm for the coordinates partitioning is needed to obtain the best performance. In the paper, the optimized partitioning of the generalized coordinates is revisited in the context of a numerical forward dynamics of the holonomic and non-holonomic multibody systems. After a short presentation of the geometric background of the optimized coordinates partitioning, a structure of the optimally partitioned vectors is discussed on the basis of a gradient analysis of the separate constraint sub-manifolds at the configuration and the velocity levels when holonomic and non-holonomic constraints are present in the system. It is shown that, for holonomic systems, the vectors of optimally partitioned coordinates have the same structure for the generalized positions and velocities. On the contrary, in the case of non-holonomic systems, the optimally partitioned coordinates generally differ at the configuration and the velocity levels. The conclusions of the paper are illustrated within the framework of the presented numerical example.     

Geometric properties of projective constraint violation stabilization method for generally constrained multibody systems on manifolds

During numerical forward dynamics of constrained multibody systems, a numerical violation of system kinematical constraints is the important issue that has to be properly treated. In this paper, the stabilized time-integration procedure, whose constraint stabilization step is based on the projection of integration results to underlying constraint manifold via post-integration correction of the selected coordinates is discussed. A selection of the coordinates is based on the optimization algorithm for coordinates partitioning. After discussing geometric background of the optimization algorithm, new formulae for optimized partitioning of the generalized coordinates are derived. Beside in the framework of the proposed stabilization algorithm, the new formulae can be used for other integration applications where coordinates partitioning is needed. Holonomic and non-holonomic systems are analyzed and optimal partitioning at the position and velocity level are considered further. By comparing the proposed stabilization method to other projective algorithms reported in the literature, the geometric and stabilization issues of the method are addressed. A numerical example that illustrates application of the method to constraint violation stabilization of non-holonomic multibody system is reported. An erratum to this article can be found at