As an important issue, we first address in this paper the problem of a regression matrix with non-homogeneous physical units in the parameter estimation of multibody systems. This matrix contains the most important information about the motion of the system. Attention must be paid to this before implementing any parameter estimation algorithm due to the unit inconsistency in matrix multiplication required in such algorithms. A procedure to treat this problem in a proper way will be proposed. An experiment on a six degrees of freedom (DOF) robotic device, whose reference link follows a desired trajectory, is performed. The data collected from the experiment are then used for sensitivity analysis of inertial parameters based on the unit-homogenized regression matrix of the system. In this way, we characterize the influence of each selected inertial parameter on the dynamics of the system using unit-consistent mathematical manipulations.
Parameter estimation is of great importance in analysis and design of multibody systems. In this context, various numerical approaches have been used to identify inertial parameters based on measurement data collected from experiments, e.g. [8], [15] and [16]. The majority of contributions devoted to this subject have focused more on some specific robotic applications, e.g. [2], [6], [7], [9], [10], [18] and [19]. Compared to the area of robotics, relatively little attention has been paid to the parameter estimation in the general area of multibody systems [3], [14] and [17]. However, this is an area of utmost importance since accurate knowledge of the inertial parameters is required for realistic dynamic simulations and analyses.
The modeling of multibody systems usually involves the development of dynamic equations. The structure of the model and the equations of motion can usually be determined but the inertial parameters are not necessarily known in advance. Standard inertial parameters associated with body ii consist of mass mimi, first moments of inertia ci=[cxi,cyi,czi]Tci=[cxi,cyi,czi]T, and moments and products of inertia JVi=[Jxxi,Jyyi,Jzzi,Jxyi,Jxzi,Jyzi]TJVi=[Jxxi,Jyyi,Jzzi,Jxyi,Jxzi,Jyzi]T with units of kg, kg m and kg m22, respectively. The elements of vectors cici and JViJVi are defined with respect to the reference frame of body ii with coordinates xx, yy and zz. With proper selection of the location of the origin of this body reference frame, the dynamic equations will be linear in terms of these inertial parameters. For parameter estimation algorithms, the system of equations of motion is first generated and then, usually reformulated in terms of the inertial parameters. Consider a system of nn bodies with total ff degrees of freedom (DOF) and rr inertial parameters (r=10nr=10n for a general system). The minimal form of the dynamic equations can be formulated and written as, [8] and [17],
equation(1)
View the MathML sourceY¯x=Q¯,
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where View the MathML sourceY¯ is an f×rf×r matrix which contains the coefficient expressions associated with the inertial parameters in the dynamic equations. The components of View the MathML sourceY¯ are functions of the position, velocity, and acceleration of the bodies. Vector xx with dimension r×1r×1 is the vector of actual inertial parameters which basically collects the standard inertial parameters of the bodies of the system, and View the MathML sourceQ¯ is the vector of applied forces and/or torques. Then, the system is moved to follow a certain trajectory and the position, velocity, and acceleration data are measured at some points. The set of identification equations for hh measurement points can be written as
equation(2)
View the MathML sourceAx=τwithA=[Y¯1T…Y¯hT]T,τ=[Q¯1T,…,Q¯hT]T,
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where AA is the regression matrix of dimension hf×rhf×r, whose elements are all known quantities, and ττ is the hf×1hf×1 vector of applied forces and/or torques determined at the measurement points. The number of measurement points should be large enough to ensure a reliable parameter estimation procedure [9] and [10].
Parameter estimation algorithms are basically based on mathematical manipulations of arrays. Such operations are required in many numerical methods (e.g. singular value decomposition (SVD), QR). If the elements of AA have different physical units, then the evaluation of such matrix products would involve the addition of terms with different units. This, however, physically makes no sense. Consequently, the identified inertial parameters, as the result of these approaches, may not be reliable and meaningful. In this paper, we propose our procedure to homogenize the units of the regression matrix of a system in order to avoid the addition of terms with different units. Then, a sensitivity analysis of a dual-pantograph device is performed in terms of some inertial parameters. We assess the effect of various inertial parameters on the dynamics based on the unit-homogenized regression matrix of the system. This can be of great importance in the analysis and design of mechanical systems. Because of the kinematic topology of a mechanical system, the inertial parameters influence the dynamics of the system with different levels of contribution [4] and [14]. Based on a sensitivity analysis, we investigate how the dynamics of a system is influenced with respect to the variation of inertial parameters. Those parameters which show no contribution may be neglected during the analysis and design step. This helps the analyst to pay more attention to the inertial parameters that have greater effect.
In this paper, we first addressed the issue of a regression matrix with non-homogeneous units in the identification model of multibody systems for the estimation of inertial parameters and sensitivity analyses. This is a very important issue from the physical point of view, which has not really been considered so far in numerical approaches developed in this context. We proposed a procedure to treat this problem in a proper way. This approach leads to a formulation where the mathematical manipulations of arrays are in complete agreement with the physical units of the elements. A key idea relates to the introduction of dimensionless inertial parameters via a linear transformation relying on eigenvectors obtained from unit-homogeneous sub-matrices of the regression matrix. This formulation makes it possible to introduce a set of essential inertial parameters in a physically consistent way. These can be used to characterize the dynamics of the system, and also to perform sensitivity studies in terms of the inertial parameters. An experimental, six DOF dual-pantograph system was used to illustrate these possibilities.
