2.1. Least squares

Among the parameter estimation methods, least squares is widely used in the identification of manipulator robots (^{Bona & Curatella, 2005}), since it is based on the use of a linear model inverse with respect to the parameters. This method allows estimating base inertial parameters from the measurement or estimation of torques and joint positions, optimizing the root mean square error of the model under the assumption that measurement errors are negligible. However, one problem in the application of this method is the noise sensitivity present in the measured data (^{Kessman, 2011}), because noise will limit the accuracy of the obtained parameters as well as the convergence rate of the algorithm. To overcome this problem, it is suggested to generate excitation trajectories or to apply a filter to data (^{Ha, Ko, & Kwon, 1989}).

The dynamic model may be written as a set of unknown parameters linear equations, as shown in Eq. (1):

where τ corresponds to the generalized forces vector, q to the generalized coordinates vector, λ to the vector or unknown dynamic parameters and Φ to the observation matrix or regres-sor. By applying the identification model to the model shown in (1), the linear equation system in λ is obtained as:

where ρ is the residual error vector. The estimated λ^ value may be obtained as:

which is the same as calculating:

2.2. Adaline neural networks

A neural network consists of a set of simple processing units, which communicate by sending signals to one another through weighted connection. Adaline (adaptive linear element) is one type of neural network. For a one-layer network with one output and one linear activation function, the output is given by Eq. (5):

This network is able to represent a linear relation between the values of the output and input units. The purpose of this network is to produce a given value y = d^p in the output when the set of values xip, i = 0, 1, ..., n. is applied to the inputs. It is intended to determine the coefficients (weights) w_i, i = 0, 1, ...,n. so the input-output response is correct for a large number of arbitrarily chosen signal sets. In this case, the "delta rule" to adjust weights is used for Adaline, training the network in such a way that the best adjustment for each set of data composed of the input values xip and desired outputs d^p is obtained. For each input datum, the network output differs from the desired value by d^p- y^p, where y^p corresponds to the current output for this pattern. The "delta rule" uses a cost or error function based on this difference to adjust the weights. The error function is given by the mean squared error. Therefore, the total error E is defined

where the p index extends over the set of input patterns and E^p represents the error over the p pattern. This procedure allows for finding the value of all weights that minimize the error function by means of the gradient descent method:

where γ is a constant of proportionality and, also:

Due to the linear relationship showed in (5):

And

Finally, we obtain:

where δ^p= (d^p- y^p) is the difference between the desired output and the current output for the p pattern.

2.3. Hopfield neural networks

Hopfield neural networks or HNNs can be used for estimating parameters on-line. In Hopfield's formulation, the dynamic of neuron i is described by the ordinary differential Equation (12) (^{Alonso, Mendonca, & Rocha, 2009}):

where u_i is the neuron input i, f_i is a strictly increasing, bounded, nonlinear continuous function, and C_i, R_i, W_ij and I_i are parameters that correspond to a capacitance, a resistance, the weight associated to the connection of the j neuron with the i neuron, and the i neuron external input or bias. In Abe's formulation, it is assumed that C_i =1 and R_i = ∞.

The considered system is linear with respect to the parameters, i.e. ithas the form shown in Eq. (1), which (1) works under the following assumptions:

Given Abe's formulation, considering a network with N neurons, u_i denotes the total input of the i neuron, its derivative with respect to time is:

where S_j represents the state (or output) of the j neuron. The input-state relation for the i neuron is given by:

where α, β>0 and, thus, ∀i e N, t ≥ t₀ S_i(t) ∈ ]-α, α[. Using matrix notation, the network is as follows:

where S(t), I(t) ϵ ℝ^N×1 and W ϵ ℝ^N×N, under the statespace representation is:

where

This a nonlinear and non-autonomous dynamic system whose architecture is completely determined by the N number of neurons, with characteristics given by α, β, W and I.

This work considers a HNN whose state S(t) over time t is taking as the estimation λ(t) of the λ parameter, in other words, λ(t) = S(t); implicitly, there is a network with as many neurons as parameters to estimate. Then, taking α = c,Eq. (14) guarantees that the trajectory generated for the network is in a feasible region of the estimation problem. Therefore, from the representation of states in Eq. (17), it is considered that: W = Φ(t)^TΦ(t) and I= Φ(t)^Tτ. In this way, the dynamic of the HHN to be used as an on-line estimation algorithm is defined by:

The advantage of using HNN to resolve the estimation problem is that inverting W = Φ(t)^TΦ(t), is not necessary, since the latter might be ill-conditioned. In addition, given its structure, prior knowledge is not required for selecting λ(t₀), i.e. of the initial estimation of λ.

Finally, the λ parameter is a globally, uniformly and asymptotically stable balance point of this HNN if the following sustains:

2.4. Extended Kalman filter

The extended Kalman filter is based on the direct dynamic model, which is nonlinear with respect to state and parameters. In this algorithm, physical parameters are considered under the use of an extended state. Kalman filter intends to estimate the x ϵ ℝⁿ state of a process controlled in discrete time (^{Welch & Bishop, 2006}). The algorithm has two steps that are executed in an iterative way, namely prediction, and correction or update of the state. However, when a system has non-linear characteristics, the formulation should be modified. This change in the algorithm is known as extended Kalman filter. In this case, the process will be represented by the following non-linear equations:

where f corresponds to a non-linear function that allows for knowing the relation between the previous step state vector k−1 and the u input with the current state vector k, according to the following procedure:

where F_k-1 and H_k corresponds the Jacobian matrices described in Eqs. (28) and (29), respectively:

where Q and R are the process and observation noise covariance matrices, respectively. In addition, they are diagonal and symmetric. On the other hand, x^k- ∈ Rn are the states estimated a priori at k instant, given the process behavior and x^k ∈ IRn the a posteriori estimated state at k instant, given the measure Z_k.

4.1. Least squares

The least squares procedure to estimate the parameters of each link of the redundant manipulator robot might be summarized in the following steps:

There is a number of methodologies to determine both the parameters influencing the systems dynamic and the set of minimum parameters required to identify the models ofsuch systems (^{Choi, Yoon, Park, & Kim, 2011}; ^{Mayeda, Yoshida, & Ohashi, 1989}; ^{Mayeda, Yoshida, & Osuka, 1990}; ^{Radkhah, Kulic, & Croft, 2007}). This work considers a numerical method that allows for identifying the set of independent parameters of the redundant manipulator robot, based on the analysis of the kernel of the regressor.

For a redundant manipulator robot with j links, ten inertial parameters are considered per link:

where m_j, p_g,j = (x_g,j y_g,j z_g,j)^T and υ_j = (I_xx,j I_yy,j I_zz,j I_xy,j I_xz,,j I_yz,j)^T_, are the mass, the position of the center of gravity, and the vector with the elements of the inertia tensor of the j link, respectively. A linear form may be composed by grouping the parameters as follows:

where

The regressor Φ is not always linearly independent or has a full rank, thus there might not be a unique solution. Nevertheless, it is possible to reduce the number of unknown parameters and transform the system into: τ = Φ^* λ^*, where λ^* is the vector for regrouped parameters. It is hard to find the linear relation between dependent and independent columns of the regressor, since dynamic relationships are complex. To find this relation, the kernel of the matrix is used, which denotes the relation between columns dependent and independent from the regressor in linear combinations. In this way, the new grouped parameters λ^* and the full rank matrix Φ^* may be found. The calculation is conducted in a recursive way, starting from the n joint up to the articulation 1.

In this case, the vector of regrouped parameters λ^* is composed of eight elements to be identified, which are presented in Table 3. The structure of the regressor Φ^* ∈ ℝ^5×8 is described in Annex B.

4.1.2. Trajectory parameterization

The prismaticjoints motion does not participate in the rotary joint dynamics. Thus, without loss of generality, trajectory optimization is only applied to rotaryjoints, since they are in conflict with the parameter estimation of least squares. The reverse is true for the prismatic link with linear characteristics. The excitation trajectories are planned using a polynomial of degree five (^{Benimeli, 2005}) in the form of Eq. (39) and the elements to be optimized are their coefficients: a_0i , a_1i , . . ., a_5i.

qi=a01+a1it+a2it2+a3it3+a4it4+a5it5 (39)

Reference sinusoidal trajectories are applied to the first and fifth link (prismatic joints). All the reference and output trajectories of all the joints of the redundant manipulator robot are shown in Figure 3. An electronic noise is observed due to the sensing (measuring) of the output trajectories.

Fig. 3 Reference and output trajectories of the redundant manipulator robot with 5 DOL. 

4.2. Adaline neural nefworks

Applying the linear model formulation to the parameters τ = Φ^* ·λ^* (see Annex B), we have that:

Therefore, when using an Adaline network, the parameters to be identified coincide with the network weights. Below, one-layer neural networks are designed with their inputs corresponding to the elements of the regressor and their outputs to the elements of torque/forces. Figure 4 shows the scheme of the networks used for the identification of the parameters of the redundant manipulator robot.

Fig. 4 Adaline neural network for identification. (a) Network 1. (b) Network 2. 

4.3. Hopfield neural nefworks

For satisfactorily identifying the parameters of the redundant manipulator robot, all intervals of t ⊂ [0, + ∞] shouldsatisfy the condition given by Eq. (20). Figure 5 shows the response produced by the HNN. The last estimated valued is used for the comparison.

Fig. 5 Online estimation of Hopfield neural networks. 

4.4. Extended Kalman filter

In this method, use is made of the dynamic model of the redundant manipulator robot, i.e. the joint coordinates that mathematically depended on the force/torque applied to them are used. The direct dynamic model is symbolically calculated from the Lagrange-Euler formulation and the grouped parameters described in Table 3, and then is expressed in the state variables presented in Eq. (41). The parameters to be identified are included in an extended state of the algorithm with a time derivative of null values, as shown in Eq. (42):

Euler approximation is used for the equations whose time derivative was calculated, as seen in Eq. (43):

Therefore, from the equation above, x^k+¹= x^k+ Δt·x˙^k is obtained, whose sampling time is At = 0.02. In turn, the initial state vectors used are initial error covariance matrix, process noise covariance matrix and observation covariance matrix, as presented in Eq.(44):

The reference trajectories used in this identification process are the same applied to the least square methods, i.e. those presented in Figure 3. The evolution of the parameter estimation curves is shown in Figure 6. In this case, the last value of the estimation is considered for error comparison.

Fig. 6 Evolution of the parameters estimated by extended Kalman filter. 

5.1. Trajectory validation

To validate the identified parameters, they are used in performance simulations on the redundant manipulator robot. In these simulations, validation trajectories different from those used in the estimation are imposed to the robot. To relate the trajectories performed by the redundant manipulator robot with the estimated parameters of the same, the following error indexes are calculated: Residual Mean Square (RMS), Residual Standard Deviation (RSD) and Agreement Index (AI); which are mathematically represented in Eqs. (45)-(47), respectively. RMS and RSD correspond to normalized percentage values between 0 and 1; indexes that explain the dispersion and deviation of the series, respectively. In addition, the AI index indicates the trend of the two series to be compared; the values that adopts may fluctuate between 0 and 1, where: 1 corresponds to a perfect match and 0 to absence of agreement. The trajectories performed by the redundant manipulator robot are shown in Figure 7, and their error indexes in Table 5. It should be noted that the trajectories presented in Figure 7 are basically overlapped, and that the values of the parameters do not show high sensitivity to following imposed trajectories:

where o_i are the values observed, o_m the mean value of the observations, p_i are the predicted values and n is the total number of observations.

Fig. 7 Validation trajectories. 

To determine the sensitivity of the identified model, due to the variations of the estimated parameters, the redundant manipulator robot is simulated, considering that its parameters varied in a restricted range of values. Also, its RMS error index is calculated between the trajectories described. Table 6 presents the range of values for the variations in the 士50% parameters. In Figure 8, the evolution of the RMS index of the performed trajectories, with respect to each estimated parameter, is illustrated.

Fig. 8 Sensitivity of the model of the redundant manipulator robot.