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Abstract

This paper presents a novel method for identifying dynamic parameters of robot manipulators with elastic joints. A procedure based on the Lagrangianm formulation of the dynamic model is proposed. Each term is inspected to search for a linear relationship with the dynamic parameters, thus enabling the linearization of robot dynamic model. Hence, the torque vector is expressed as the product of a regressor matrix, suitably defined by the vector of dynamic parameters. A parametric identification based on a least-squares technique is applied to determine dynamic parameters of robots with elastic joints. The correctness of the proposed procedure has been tested in simulation on two robotic structures with elastic joints of different complexity, that is, a 2-degree-of-freedom (dof) and a 6-dof manipulator, controlled with a PD control in the joint space.

1. Introduction

Robot dynamics describes the relationship between motion and forces by means of a number of parameters, including link and motor masses, inertias, and friction. Dynamic parameters can be grouped into the following three main categories, based on the contribution they provide to generate motor torques [1]:

unidentifiable parameters: if a variation of the parameter does not modify the robot dynamic behavior;

linearly independent parameters: if a variation of the parameter modifies the robot dynamic behavior in a way that is not repeatable by varying another parameter (or a set of other parameters);

linearly dependent parameters: if a variation of the parameter modifies the robot dynamic behavior in a way that is repeatable by varying another parameter (or a set of other parameters).

The knowledge of robot dynamic parameters is helpful for simulation purposes [2], for control purposes [3, 4], and for mechatronic design purposes (i.e., to optimize robot design by studying the interaction between the robot and its environment [5]). Notwithstanding, a proper estimation is often unavailable. Hence, the need emerges of resorting to methods for estimating them; for clarity, a schematic classification of these methods is provided in Figure 1, with a brief description.

Figure 1:

Methods for dynamic parameters identification (readapted from [26]).

This paper is focused on one specific branch of the tree in Figure 1, that is, the Parameters identification procedures belonging to the category of offline estimation.

They consist of “exciting” the dynamics of a manipulator making it move along optimized trajectories through proper torque commands [1, 6]. These methods are generally used to identify parameters of open kinematic chains, generally neglecting coupled dynamics and transmission elasticity. They are grounded on three possible approaches. The first approach exploits the Newton-Euler formulation (N.-E.) of the dynamic model [1, 7, 8]. The second approach is based on the Lagrangian formulation of the dynamic model [9, 10]. The third one exploits again energetic considerations [11, 12], as in the second approach based on the Lagrangian formulation. However, it evaluates the energy of each configuration without relating the external torques to the Lagrangian of the system. In [13] a comparative analysis of the computational burden of the Lagrangian formulation with respect to the energetic formulation is provided.

All the aforementioned approaches are formulated for robots with rigid joints and links and cannot be easily extended to the case of robots with mechanical elasticity because of the different dynamic model. On the other hand, when existing, elastic phenomena cannot be neglected because of the degradation of robot performance they may cause [3, 14, 15].

It is worth noticing that only a few works cope with the identification of the elasticity in single joints due to the transmission systems and, to our knowledge, no work does exist on the identification of the complete dynamics of robots with mechanical elasticity. For instance, in [16] a method to estimate one actuator elasticity is proposed, but motor inertia is supposed to be known. The work in [17] presents the parameter identification of one elastic joint based on a nonlinear model, and at least four series of experiments are needed to apply the proposed approach.

This paper intends to propose a novel procedure for identifying the dynamic parameters of robots with elastic joints. It is grounded on the Lagrange formulation of the dynamic model of the manipulator, thus accounting for robot elastic energy in addition to kinetic and gravitational energy contributions. Open kinematic chains and absence of coupled dynamics are supposed. Furthermore, joint elasticity is assumed to be concentrated in the transmission system between the motor and the joint. This entails that 14 dynamic parameters should be estimated for each joint. They are link mass, three coordinates of the center of mass, six components of the inertia tensor, motor inertia, static and viscous friction coefficients (assuming the Coulomb friction model for static friction), and transmission elasticity. Special attention is paid to the extraction of the regressor matrix, in order to linearize robot dynamic model and facilitate parametric identification. Also a method is proposed to identify the category to which a parameter belongs (i.e., unidentifiable, linearly dependent, or else independent).

The paper is structured as follows. Section 2 introduces the notation that is adopted in this paper. In Section 3, linearization of robot dynamic model with respect to dynamic parameters is presented. The regressor matrix relating external and actuation torques with dynamic parameters is extracted in Section 4. Parameter identification is carried out in Section 5; it also proposes a method to discriminate the category to which a parameter belongs. Finally, the proposed procedure is validated on two simulated robots (a planar and a 6-dof robot) in Section 6. Section 7 summarizes the main achievements of this work, presenting conclusions and future works.

2. Notation

In this paper, the following notation is used. All the parameters are defined in the robot base frame, that is, “0” frame, if not specified by other apexes. Regarding kinematic parameters we have the following:

θ_i: angular position of motor i, divided by reduction gear $k_{m_{i}}$ of transmission i;

q_i: angular position of link i, which is different from θ_i due to transmission elasticity;

p_i, ${\dot{p}}_{i}$ , R_i, ω_i: position, velocity, rotation matrix, and angular velocity of reference frame i on link i;

$z_{m_{i}}$ : rotation axis of rotor of motor i, expressed in frame i − 1.

For defining robot dynamic parameters, it is useful to introduce the concept of “body.” Body i is the system composed of link i and the stator of the motor rigidly connected to it. Thus, the following notation applies to dynamic parameters:

m_i: mass of body i;

c_i: product of mass and center of mass of body i, represented in frame i;

I_i: inertia tensor of body i represented in frame i and relatively to its frame origin;

Υ_i: inertia of rotor of motor i;

K_i: elasticity of transmission i;

f_si, f_vi: static and viscous friction coefficients of motor i. Coulomb friction model is assumed. Friction is supposed to be concentrated in motors, as proposed in the literature [18, 19]. In other words, it is assumed that all the components of link i prone to friction rotate with angular velocity θ_i.

3. Dynamic Model of Robot Manipulators with Elastic Joints

The following assumptions are provided:

The robot has an open kinematic chain of rigid bodies, driven by electrical actuators through elastic joints undergoing small deformations in the domain of linear elasticity.

Rotors of motors are rigid bodies with uniform density around their rotation axes.

Stator of motor i is rigidly connected to joint i − 1.

Robot kinetic energy T, gravitational potential energy U_g, and elastic potential energy U_e can be computed as follows:

\begin{matrix} T = \sum_{i = 1}^{n} ‍ (\frac{1}{2} m_{i} {\dot{p}}_{i}^{T} {\dot{p}}_{i} + {\dot{p}}_{i}^{T} ω_{i} \times (R_{i} c_{i}) + \frac{1}{2} ω_{i}^{T} R_{i} I_{i} R_{i}^{T} ω_{i} + k_{m_{i}} {\dot{θ}}_{i} Υ_{i} z_{m_{i}}^{T} R_{i - 1}^{T} ω_{i - 1} + \frac{1}{2} k_{m_{i}}^{2} {\dot{θ}}_{i}^{2} Υ_{i}), \\ U_{g} = - \sum_{i = 1}^{n} ‍ (m_{i} g_{0}^{T} p_{i} + g_{0}^{T} R_{i} c_{i}), \\ U_{e} = \sum_{i = 1}^{n} ‍ K_{i} {(q_{i} - θ_{i})}^{2}, \end{matrix}

(1)

where n is the number of robot joints and g₀ represents the gravity vector. For the analytical formulation of T, refer to [20] and Appendix A; for U_g, refer to [20]; for U_e, refer to [21].

By applying the Lagrangian formulation, dynamic model of robots with elastic joints can be expressed as [3, 21]

τ = B (q) \ddot{\bar{q}} + C (q, \dot{\bar{q}}) \dot{\bar{q}} + f_{v} \dot{\bar{q}} + f_{s} sign (\dot{\bar{q}}) + g (q) + K_{e} \bar{q},

(2)

where $τ^{T} = {[{(J {(q)}^{T} h_{e})}^{T} u^{T}]}^{T} \in R^{(2 n \times 1)}$ is the external torque vector, $J (q) \in R^{(6 \times n)}$ is the robot Jacobian matrix, $h_{e} \in R^{(6 \times 1)}$ is the generalized vector of external forces, and $u \in R^{(n \times 1)}$ is the actuation torque vector; $\bar{q} = {[\begin{bmatrix} q^{T} & θ^{T} \end{bmatrix}]}^{T} \in R^{(2 n \times 1)}$ is the generalized joint vector, and

$B (q) \in R^{(2 n \times 2 n)}$ is the inertia matrix, related to total robot kinetic energy T as follows:

T = \frac{1}{2} {\dot{\bar{q}}}^{T} B (q) \dot{\bar{q}};

(3)

matrix $C (q, \dot{\bar{q}}) \in R^{(2 n \times 2 n)}$ is the centrifugal and Coriolis torque matrix, related to elements of B(q) as

C (q, \dot{\bar{q}}) = \sum_{k = 1}^{n} ‍ {\dot{q}}_{k} H^{(k)} (q) - asym ([\begin{bmatrix} {\dot{\bar{q}}}^{T} H^{(1)} (q) \\ ⋮ \\ {\dot{\bar{q}}}^{T} H^{(n)} (q) \\ zeros (n, 2 n) \end{bmatrix}]),

(4)

being operator asym(M) defined as asym(M) = (M − M^T)/2 and H^(k)(q) = ∂B(q)/∂q_k; zeros(n, 2n) represents the (n × 2n) null matrix. The relationship between matrix $C (q, \dot{\bar{q}})$ and matrices H^(k)(q), k = 1…n, is shown in Appendix B;

the generalized static and viscous friction matrices f_s and f_v are given by

\begin{matrix} f_{s} = [\begin{bmatrix} zeros (n, n) & zeros (n, n) \\ zeros (n, n) & diag {f_{s_{1}}, \dots, f_{s_{n}}} \end{bmatrix}], \\ f_{v} = [\begin{bmatrix} zeros (n, n) & zeros (n, n) \\ zeros (n, n) & diag {f_{v_{1}}, \dots, f_{v_{n}}} \end{bmatrix}]; \end{matrix}

(5)

g(q) = ∂U_g/∂q is the gravity vector, where U_g is the gravitational potential energy;

$K_{e} \bar{q} = \partial U_{e} / \partial \bar{q}$ is the contribution of elasticity, being K_e expressed as [21]

K_{e} = [\begin{bmatrix} K & - K \\ - K & K \end{bmatrix}],

(6)

where U_e is the elastic potential energy and K = diag ⁡ {K₁, …, K_n} is the joints elasticity matrix.

As shown in Appendix A, by exploiting the relationships between linear and angular velocity with joint velocity, inertia matrix B(q) can be written as follows:

B (q) = [\begin{bmatrix} B_{m} (q) + B_{c} (q) + B_{I} (q) & B_{d} (q) \\ B_{d} {(q)}^{T} & B_{Υ} \end{bmatrix}],

(7)

where

\begin{matrix} B_{m} (q) = \sum_{i = 1}^{n} ‍ m_{i} J_{P}^{{(i)}^{T}} J_{P}^{(i)}, \\ B_{c} (q) = \sum_{i = 1}^{n} ‍ (J_{O}^{{(i)}^{T}} {(R_{i} c_{i})}_{[x]} J_{P}^{(i)} - J_{P}^{{(i)}^{T}} {(R_{i} c_{i})}_{[x]} J_{O}^{(i)}), \\ B_{I} (q) = \sum_{i = 1}^{n} ‍ J_{O}^{{(i)}^{T}} R_{i} I_{i} R_{i}^{T} J_{O}^{(i)}, \\ B_{d} (q) = [\begin{bmatrix} Υ_{1} k_{m_{1}} J_{O}^{{(0)}^{T}} & R_{0} z_{m_{1}} & \dots & Υ_{n} k_{m_{n}} J_{O}^{{(n - 1)}^{T}} & R_{n - 1} z_{m_{n}} \end{bmatrix}], \\ B_{Υ} = diag \{Υ_{1} k_{m_{1}}^{2}, \dots, Υ_{n} k_{m_{n}}^{2}\} . \end{matrix}

(8)

Symbol (·)_[x] denotes the skew-symmetric matrix corresponding to the cross product. Also note that (·)_[x]^T = − (·)_[x]. J_P⁽ⁱ⁾ and J_O⁽ⁱ⁾ are the position and orientation Jacobians of body i.

It is worth observing that submatrix B_m(q) is linear with body masses m_i, B_c(q) is linear with products of body masses and body centers of mass c_i, B_I(q) is linear with body inertia tensors I_i, and submatrices B_d(q) and B_Υ are linear with rotor inertias Υ_i.

Once matrix B(q) is defined, matrices H^(k)(q) can be easily evaluated as

H^{(k)} (q) = [\begin{bmatrix} H_{m}^{(k)} (q) + H_{c}^{(k)} (q) + H_{I}^{(k)} (q) & H_{d}^{(k)} (q) \\ H_{d}^{(k)} {(q)}^{T} & zeros (n, n) \end{bmatrix}],

(9)

where

\begin{matrix} H_{m}^{(k)} (q) = \sum_{i = 1}^{n} ‍ m_{i} (A_{P, (k)}^{{(i)}^{T}} J_{P}^{(i)} + J_{P}^{{(i)}^{T}} A_{P, (k)}^{(i)}), \\ H_{c}^{(k)} (q) = \sum_{i = 1}^{n} ‍ (A_{O, (k)}^{{(i)}^{T}} {(R_{i} c_{i})}_{[x]} J_{P}^{(i)} + J_{O}^{{(i)}^{T}} {({(J_{O, k}^{(i)})}_{[x]} R_{i} c_{i})}_{[x]} J_{P}^{(i)} + J_{O}^{{(i)}^{T}} {(R_{i} c_{i})}_{[x]} A_{P, (k)}^{(i)} - A_{P, (k)}^{{(i)}^{T}} {(R_{i} c_{i})}_{[x]} J_{O}^{(i)} - J_{P}^{{(i)}^{T}} {({(J_{O, k}^{(i)})}_{[x]} R_{i} c_{i})}_{[x]} J_{O}^{(i)} - J_{P}^{{(i)}^{T}} {(R_{i} c_{i})}_{[x]} A_{O, (k)}^{(i)}), \\ H_{I}^{(k)} (q) = \sum_{i = 1}^{n} ‍ (A_{O, (k)}^{{(i)}^{T}} R_{i} I_{i} R_{i}^{T} J_{O}^{(i)} + J_{O}^{{(i)}^{T}} {(J_{O, k}^{(i)})}_{[x]} R_{i} I_{i} R_{i}^{T} J_{O}^{(i)} - J_{O}^{{(i)}^{T}} R_{i} I_{i} R_{i}^{T} {(J_{O, k}^{(i)})}_{[x]} J_{O}^{(i)} + J_{O}^{{(i)}^{T}} R_{i} I_{i} R_{i}^{T} A_{O, (k)}^{(i)}) \\ H_{d}^{(k)} (q) = [Υ_{1} r_{a_{0}}^{(k)} \dots Υ_{n} r_{a_{n}}^{(k)}], \end{matrix}

(10)

being

r_{a_{i}}^{(k)} = k_{m_{i}} (A_{O, (k)}^{{(i - 1)}^{T}} R_{i - 1} + J_{O}^{{(i - 1)}^{T}} {(J_{O, k}^{(i - 1)})}_{[x]} R_{i - 1}) z_{m_{i}},

(11)

and A_{P, (k)}⁽ⁱ⁾ = ∂J_P⁽ⁱ⁾/∂q_k, A_{O, (k)}⁽ⁱ⁾ = ∂J_O⁽ⁱ⁾/∂q_k, and (J_{O, k}⁽ⁱ⁾)_[x]R_i = ∂R_i/∂q_k.

Finally, coefficient j of gravity vector g(q) can be easily shown to be equal to

g {(q)}_{j} = \frac{\partial U_{g}}{\partial q_{j}} = \{\begin{cases} g {(q)}_{m, j} + g {(q)}_{c, j} & j \leq n \\ 0 & j > n, \end{cases}

(12)

where g(q)_{m, j} = − ∑^{i = 1}_n‍m_ig₀^TJ_{P, j}⁽ⁱ⁾ depends only on masses, while g(q)_{c, j} = − ∑^{i = 1}_n‍g₀^T(J_{O, j}⁽ⁱ⁾)_[x]R_ic_i depends only on products of masses and centers of gravity.

4. Linearization of Robot Dynamic Model

Formulation of robot dynamics as in Section 3 makes each term depend only on one dynamic parameter. This has the consequent advantage of simplifying the linearization of the dynamic model and the evaluation of regressor matrix $Y (\bar{q}, \dot{\bar{q}}, \ddot{\bar{q}})$ , which relates the measured generalized external torques τ to the robot dynamic parameters. Hence, robot dynamic model in (2) can be linearized as

τ = (Y_{B} (q, \ddot{\bar{q}}) + Y_{C} (q, \dot{\bar{q}}) + Y_{oth} (\bar{q}, \dot{θ})) Π = Y (\bar{q}, \dot{\bar{q}}, \ddot{\bar{q}}) Π,

(13)

where $Y_{B} (q, \ddot{\bar{q}})$ is the regressor of the inertia matrix, $Y_{C} (q, \dot{\bar{q}})$ is the regressor of the centrifugal and Coriolis torque matrix, $Y_{oth} (\bar{q}, \dot{θ})$ is the regressor of all the remaining terms in (2), and

Π = {[Π_{m}^{T} Π_{c}^{T} Π_{I}^{T} Π_{Υ}^{T} Π_{f_{s}}^{T} Π_{f_{v}}^{T} Π_{K}^{T}]}^{T}

(14)

is the parameter vector composed of the subvectors of mass parameters Π_m = [m₁ ⋯ m_n]^T, product of mass and center of mass parameters Π_c = [c₁^T ⋯ c_n^T]^T, body inertia parameters Π_I = [I₁^*T ⋯ I_n^*T]^T, rotor inertia parameters Π_Υ = [Υ₁, …, Υ_n]^T, static friction coefficient parameters $Π_{f_{s}} = {[f_{s, 1} \dots f_{s, n}]}^{T}$ , viscous friction coefficient parameters $Π_{f_{v}} = {[f_{v, 1} \dots f_{v, n}]}^{T}$ , and elasticity parameters Π_K = [K₁ ⋯ K_n]^T. Body inertia I_i^* is expressed as I_i^* = [I_{xx, i} I_{yy, i} I_{zz, i} I_{xy, i} I_{xz, i} I_{yz, i}]^T, where the first three parameters represent the inertia moments and the last three parameters represent the inertia products.

In the following the expression of the regressor matrix related to each term of the dynamic model is presented.

First, product $B (q) \ddot{\bar{q}}$ is considered. It can be rewritten as

B (q) \ddot{\bar{q}} = [\begin{bmatrix} Y_{B_{m}} (q, \ddot{q}) Π_{m} + Y_{B_{c}} (q, \ddot{q}) Π_{c} + Y_{B_{I}} (q, \ddot{q}) Π_{I} + Y_{B_{Υ 1}} (q, \ddot{θ}) Π_{Υ} \\ Y_{B_{Υ 2}} (q, \ddot{\bar{q}}) Π_{Υ} \end{bmatrix}] .

(15)

The regressor matrices in (15) can be obtained from (8) as 1

Y_{B_{m}} (q, \ddot{q}) = [\begin{bmatrix} Y_{B_{m, 1}} & \dots & Y_{B_{m, n}} \end{bmatrix}] \in R^{n \times n} being Y_{B_{m, i}} = J_{P}^{{(i)}^{T}} J_{P}^{(i)} \ddot{q} \in R^{n \times 1}

(16)

Y_{B_{c}} (q, \ddot{q}) = [\begin{bmatrix} Y_{B_{c, 1}} & \dots & Y_{B_{c, n}} \end{bmatrix}] \in R^{n \times 3 n} being Y_{B_{c, i}} = (J_{P}^{{(i)}^{T}} {(J_{O}^{(i)} \ddot{q})}_{[x]} - J_{O}^{{(i)}^{T}} {(J_{P}^{(i)} \ddot{q})}_{[x]}) R_{i} \in R^{n \times 3}

(17)

Y_{B_{I}} (q, \ddot{q}) = [\begin{bmatrix} Y_{B_{I, 1}} & \dots & Y_{B_{I, n}} \end{bmatrix}] \in R^{n \times 6 n} being Y_{B_{I, i}} = J_{O}^{{(i)}^{T}} R_{i} {(R_{i}^{T} J_{O}^{(i)} \ddot{q})}_{[⊳ ⊲]} \in R^{n \times 6}

(18)

Y_{B_{Υ 1}} (q, \ddot{θ}) = [\begin{bmatrix} Y_{B_{Υ 1,1}} & \dots & Y_{B_{Υ 1, n}} \end{bmatrix}] \in R^{n \times n} being Y_{B_{Υ 1, i}} = {\ddot{θ}}_{i} k_{m_{i}} J_{O}^{{(i - 1)}^{T}} R_{i - 1} z_{m_{i}} \in R^{n \times 1}

(19)

Y_{B_{Υ 2}} (q, \ddot{\bar{q}}) = diag \{\begin{Bmatrix} Y_{B_{Υ 2,1}} & \dots & Y_{B_{Υ 2, n}} \end{Bmatrix}\} \in R^{n \times n} being Y_{B_{Υ 2, i}} = k_{m_{i}} z_{m_{i}}^{T} R_{i - 1}^{T} J_{O}^{(i - 1)} \ddot{q} + k_{m_{i}}^{2} {\ddot{θ}}_{i} \in R^{1 \times 1} .

(20)

From (16)–(20) it follows that $B_{m} \ddot{q} = Y_{B_{m}} Π_{m}$ , $B_{c} \ddot{q} = Y_{B_{c}} Π_{c}$ , $B_{I} \ddot{q} = Y_{B_{I}} Π_{I}$ , $B_{d} \ddot{θ} = Y_{B_{Υ 1}} Π_{Υ}$ , and $B_{d}^{T} \ddot{q} + B_{Υ} \ddot{θ} = Y_{B_{Υ 2}} Π_{Υ}$ .

Consequently, product $B (q) \ddot{\bar{q}}$ can be expressed as $Y_{B} (q, \ddot{\bar{q}}) Π$ , where

Y_{B} (q, \ddot{\bar{q}}) = [\begin{bmatrix} Y_{B_{m}} (q, \ddot{q}) & Y_{B_{c}} (q, \ddot{q}) & Y_{B_{I}} (q, \ddot{q}) & Y_{B_{Υ 1}} (q, \ddot{θ}) & zeros (n, 3 n) \\ zeros (n, n) & zeros (n, 3 n) & zeros (n, 6 n) & Y_{B_{Υ 2}} (q, \ddot{\bar{q}}) & zeros (n, 3 n) \end{bmatrix}] \in R^{(2 n \times 14 n)} .

(21)

Analogously, product $C (q, \dot{\bar{q}}) \dot{\bar{q}}$ can be written as (see Appendix B)

C (q, \dot{\bar{q}}) \dot{\bar{q}} = Y_{C} (q, \dot{\bar{q}}) Π = (\sum_{k = 1}^{n} ‍ ({\dot{q}}_{k} Y_{{\bar{H}}^{(k)}} (q, \dot{\bar{q}})) - \frac{1}{2} [\begin{bmatrix} {\dot{\bar{q}}}^{T} Y_{H^{(1)}} (q, \dot{\bar{q}}) \\ ⋮ \\ {\dot{\bar{q}}}^{T} Y_{H^{(n)}} (q, \dot{\bar{q}}) \\ zeros (n, 2 n) \end{bmatrix}]) Π,

(22)

where $Y_{H^{(k)}} (q, \dot{\bar{q}})$ is defined so that $H^{(k)} (q) \dot{\bar{q}} = Y_{H^{(k)}} (q, \dot{\bar{q}}) Π$ . Similarly to matrix $Y_{B} (q, \ddot{\bar{q}})$ , matrix $Y_{H^{(k)}} (q, \dot{\bar{q}})$ is given by

Y_{H^{(k)}} (q, \dot{\bar{q}}) = [\begin{bmatrix} Y_{H_{m}^{(k)}} (q, \dot{q}) & Y_{H_{c}^{(k)}} (q, \dot{q}) & Y_{H_{I}^{(k)}} (q, \dot{q}) & Y_{H_{Υ 1}^{(k)}} (q, \dot{θ}) & zeros (n, 3 n) \\ zeros (n, n) & zeros (n, 3 n) & zeros (n, 6 n) & Y_{H_{Υ 2}^{(k)}} (q, \dot{q}) & zeros (n, 3 n) \end{bmatrix}] \in R^{(2 n \times 14 n)},

(23)

where

Y_{H_{m}^{(k)}} (q, \dot{q}) = [\begin{bmatrix} Y_{H_{m, 1}^{(k)}} & \dots & Y_{H_{m, n}^{(k)}} \end{bmatrix}] \in R^{n \times n} being Y_{H_{m, i}^{(k)}} = A_{P, (k)}^{{(i)}^{T}} J_{P}^{(j)} \dot{q} + J_{P}^{{(j)}^{T}} A_{P, (k)}^{(i)} \dot{q} \in R^{n \times 1}

(24)

Y_{H_{c}^{(k)}} (q, \dot{q}) = [\begin{bmatrix} Y_{H_{c, 1}^{(k)}} & \dots & Y_{H_{c, n}^{(k)}} \end{bmatrix}] \in R^{n \times 3 n} being Y_{H_{c, i}^{(k)}} = (A_{P, (k)}^{{(i)}^{T}} {(J_{O}^{(i)} \dot{q})}_{[x]} + J_{P}^{{(i)}^{T}} {(A_{O, (k)}^{(i)} \dot{q})}_{[x]} + J_{P}^{{(i)}^{T}} {(J_{O}^{(i)} \dot{q})}_{[x]} {(J_{O, k}^{(i)})}_{[x]} - A_{O, (k)}^{{(i)}^{T}} {(J_{P}^{(i)} \dot{q})}_{[x]} - J_{O}^{{(i)}^{T}} {(A_{O, (k)}^{{(i)}^{T}} \dot{q})}_{[x]} - J_{O}^{{(i)}^{T}} {(J_{P}^{(i)} \dot{q})}_{[x]} {(J_{O, k}^{(i)})}_{[x]}) R_{i} \in R^{n \times 3},

(25)

Y_{H_{I}^{(k)}} (q, \dot{q}) = [\begin{bmatrix} Y_{H_{I, 1}^{(k)}} & \dots & Y_{H_{I, n}^{(k)}} \end{bmatrix}] \in R^{n \times 6 n} being Y_{H_{I, i}^{(k)}} = A_{O, (k)}^{{(i)}^{T}} R_{i} {(R_{i}^{T} J_{O}^{(i)} \dot{q})}_{[⊳ ⊲]} + J_{O}^{{(i)}^{T}} {(J_{O, k}^{(i)})}_{[x]} R_{i} {(R_{i}^{T} J_{O}^{(i)} \dot{q})}_{[⊳ ⊲]} - J_{O}^{{(i)}^{T}} R_{i} {(R_{i}^{T} {(J_{O, k}^{(i)})}_{[x]} J_{O}^{(i)} \dot{q})}_{[⊳ ⊲]} + J_{O}^{{(i)}^{T}} R_{i} {(R_{i}^{T} A_{O, (k)}^{(i)} \dot{q})}_{[⊳ ⊲]} \in R^{n \times 6},

(26)

Y_{H_{Υ 1}^{(k)}} (q, \dot{θ}) = [\begin{bmatrix} Y_{H_{Υ 1,1}^{(k)}} & \dots & Y_{H_{Υ 1, n}^{(k)}} \end{bmatrix}] \in R^{n \times n} being Y_{H_{Υ 1, i}^{(k)}} = {\dot{θ}}_{i} r_{a_{i}}^{(k)} \in R^{n \times 1}

(27)

Y_{H_{Υ 2}^{(k)}} (q, \dot{q}) = diag \{\begin{Bmatrix} Y_{H_{Υ 2,1}^{(k)}} & \dots & Y_{H_{Υ 2, n}^{(k)}} \end{Bmatrix}\} \in R^{n \times n} being Y_{H_{Υ, i}^{(k)}} = r_{a_{i}}^{{(k)}^{T}} \dot{q} \in R^{1 \times 1},

(28)

where vectors $r_{a_{i}}$ are defined as in (11).

As regards the remaining conservative terms, that is, $g (q) + K_{e} \bar{q}$ , and the dissipative terms, that is, $f_{v} \dot{\bar{q}} + f_{s} sign (\dot{\bar{q}})$ , they can be written as

g (q) + K_{e} \bar{q} + f_{v} \dot{\bar{q}} + f_{s} sign (\dot{\bar{q}}) = Y_{oth} (\bar{q}, \dot{θ}) Π,

(29)

where

Y_{oth} (\bar{q}, \dot{θ}) = [\begin{bmatrix} Y_{g_{m}} (q) & Y_{g_{c}} (q) & zeros (n, 7 n) & zeros (n, n) & zeros (n, n) & Y_{K} (\bar{q}) \\ zeros (n, n) & zeros (n, 3 n) & zeros (n, 7 n) & Y_{f_{s}} (\dot{θ}) & Y_{f_{v}} (\dot{θ}) & - Y_{K} (\bar{q}) \end{bmatrix}] \in R^{(2 n \times 14 n)},

(30)

being both $Y_{g_{m}} (q)$ and $Y_{g_{c}} (q)$ upper triangular matrices satisfying relations $g {(q)}_{m, j} = Y_{g_{m}} (q) Π_{m}$ and $g {(q)}_{c, j} = Y_{g_{c}} (q) Π_{c}$ with

\begin{matrix} Y_{g_{m}} (q) = [\begin{bmatrix} - g_{0}^{T} J_{P, 1}^{(1)} & \dots & - g_{0}^{T} J_{P, 1}^{(n)} \\ ⋱ & ⋮ \\ 0 & - g_{0}^{T} J_{P, n}^{(n)} \end{bmatrix}] \in R^{n \times n}, \\ Y_{g_{c}} (q) = [\begin{bmatrix} - g_{0}^{T} {(J_{O, 1}^{(1)})}_{[x]} R_{1} & \dots & - g_{0}^{T} {(J_{O, 1}^{(n)})}_{[x]} R_{n} \\ ⋱ & ⋮ \\ 0 & - g_{0}^{T} {(J_{O, n}^{(n)})}_{[x]} R_{n} \end{bmatrix}] \in R^{n \times 3 n} . \end{matrix}

(31)

On the other hand, $Y_{f_{s}}$ , $Y_{f_{v}}$ , and Y_K are diagonal matrices defined as

Y_{f_{s}} (\dot{θ}) = diag \{sign ({\dot{θ}}_{1}), \dots, sign ({\dot{θ}}_{n})\} \in R^{n \times n},

(32)

Y_{f_{v}} (\dot{θ}) = diag \{{\dot{θ}}_{1}, \dots, {\dot{θ}}_{n}\} \in R^{n \times n},

(33)

Y_{K} (\bar{q}) = diag \{q_{1} - θ_{1}, \dots, q_{n} - θ_{n}\} \in R^{n \times n} .

(34)

5. Parametric Identification Procedure

A parametric identification procedure for robots with elastic joints is proposed to identify vector of dynamic parameters Π. As for the rigid case [7, 10], it exploits the property of linearity of robot dynamic model with respect to the vector of dynamic parameters in (13). The parametric identification procedure requires recording manipulator actuation torques u, and motion variables $\bar{q}$ , $\dot{\bar{q}}$ , and $\ddot{\bar{q}}$ , while moving the robot in the free space along suitable trajectories. For each measurement k, with k = 1, …, w, matrix $W_{k} = Y ({\bar{q}}_{k}, {\dot{\bar{q}}}_{k}, {\ddot{\bar{q}}}_{k}) \in R^{2 n \times 14 n}$ and vector $T_{k} = {[\begin{bmatrix} 0^{T} & u_{k}^{T} \end{bmatrix}]}^{T} \in R^{2 n \times 1}$ can be evaluated and, by resorting to (13), the following relation can be written:

[\begin{bmatrix} T_{1} \\ ⋮ \\ T_{w} \end{bmatrix}] = [\begin{bmatrix} W_{1} \\ ⋮ \\ W_{w} \end{bmatrix}] Π,

(35)

or, alternatively,

T = W Π,

(36)

where

W = [\begin{bmatrix} W_{1} \\ ⋮ \\ W_{w} \end{bmatrix}] \in R^{(2 n w \times 14 n)}, T = [\begin{bmatrix} T_{1} \\ ⋮ \\ T_{w} \end{bmatrix}] \in R^{(2 n w \times 1)} .

(37)

If matrix W is of full rank and the number of rows is greater than (or at least equal to) the number of columns, parameter vector Π can be identified as

Π = W^{†} T,

(38)

where W^† = (W^TW)^{− 1}W^T is the left pseudoinverse matrix of W. It is worth noticing that, regarding relation (36) as a system of 2nw equations in 14n variables, at least 7 measurements are required for the identification of the vector Π; thus w ≥ 7.

Otherwise, if rank of matrix W is not full, it is possible to identify the category to which a single parameter belongs [1] with the procedure explained as follows.

If column i of matrix W is a null vector, then parameter π_i is unidentifiable and does not affect robot dynamics.

If column i of matrix W is not null, but it can be obtained as a linear combination of other columns, then parameter π_i is identifiable only in linear combination with other parameters.

If column i of matrix W is not null and cannot be obtained as a linear combination of other columns, then parameter π_i is independently identifiable.

In order to evaluate whether parameters are independently identifiable or not, Gauss-Jordan elimination procedure can be applied to evaluate a base matrix L for matrix W, that is, a set of linearly independent columns of matrix W. Then, matrix K can be introduced, composed of not-null columns of matrix W which are not columns of matrix L. Thus, each column K_i of matrix K can be written as a linear combination of columns L_j of matrix L as follows:

K_{i} = a_{i, 1} L_{1} + \dots + a_{i, m} L_{m} = L A_{i} A_{i} = {[\begin{bmatrix} a_{i, 1} & \dots & a_{i, m} \end{bmatrix}]}^{T},

(39)

where m is the number of vectors composing the base of W.

Eventually, by grouping vector Π into two subvectors Π_L and Π_K, the following relations can be obtained:

W Π = [\begin{bmatrix} L & K \end{bmatrix}] [\begin{bmatrix} Π_{L} \\ Π_{K} \end{bmatrix}],

(40)

and, due to relation (39),

W Π = L [\begin{bmatrix} I_{(m \times m)} & A \end{bmatrix}] [\begin{bmatrix} Π_{L} \\ Π_{K} \end{bmatrix}] = L Π^{*},

(41)

where I_{(m × m)} is the (m × m) identity matrix, $A = [\begin{bmatrix} A_{1} & \dots & A_{14 n - m} \end{bmatrix}]$ is the matrix composed of columns A_i, and Π^* = Π_L + AΠ_K is the vector of parameters to be identified. Thus, parameter $Π_{L_{j}}$ is independently identifiable if row j of matrix A is a null vector; otherwise, parameter $Π_{L_{j}}$ is identifiable only in linear combination with some parameters of vector Π_K. Parameters of vector Π_K are identifiable only in linear combination.

Further, vector Π^* can be computed through a least-squares approach as

Π^{*} = {(L^{T} L)}^{- 1} L^{T} T .

(42)

6. Validation and Results

Simulation tests have been performed to validate the proposed identification procedure of dynamic parameters on robots with elastic joints. To this purpose, two robotic structures of different complexity have been modeled. The former is a planar manipulator with 2 dofs and elastic joints (see Figure 2). The latter is a 6-dof manipulator modeled as a PUMA 560 with additional elastic joints.

Figure 2:

2-dof planar robot with elastic joints. Gravity vector is directed along − y₀ axis.

The planar manipulator moves in the vertical plane; hence, gravitational acceleration g₀ is directed along axis − y₀ with respect to the base reference system ( $g_{0} = {[\begin{bmatrix} 0 & - 9.81 & 0 \end{bmatrix}]}^{T} m / s^{2}$ ).

Kinematic and dynamic parameters of the planar robot are as follows: a₁ = a₂ = 0.5 m (a_i: link length), l₁ = l₂ = 0.25 m (l_i: center of mass distance; see Figure 2), m₁ = 20 kg, m₂ = 10 kg (m_i: link mass), $m_{m_{1}} = m_{m_{2}} = 1 kg$ ( $m_{m_{i}}$ : motor mass), $k_{m_{1}} = k_{m_{2}} = 0.1$ ( $k_{m_{i}}$ : transmission reduction gear), K₁ = 3000 N·m/rad, K₂ = 1800 N·m/rad (K_i: transmission elasticity), $I_{l_{1}} = 0.4667 kg \cdot m^{2}$ , $I_{l_{2}} = 0.2333 kg \cdot m^{2}$ ( $I_{l_{i}}$ : link inertia around rotation axis), Υ₁ = 21.18 kg·m², and Υ₂ = 12.10 kg·m² (Υ_i: motor inertia). It is supposed that friction torque F_i acts on joint i. It accounts for the Stribeck effect and is modeled as follows [22]:

F_{i} = f_{1, i} {\dot{θ}}_{i} + f_{2, i} sign ({\dot{θ}}_{i}) - f_{3, i} sign ({\dot{θ}}_{i}) e^{- | {\dot{θ}}_{i} | / f_{4, i}} - f_{5, i} sign ({\dot{θ}}_{i}) e^{- 1 / f_{6, i} |{\dot{θ}}_{i}|},

(43)

where

\begin{matrix} f_{1,1} = 0.1434 f_{1,2} = 0.1391 \\ f_{2,1} = 0.3302 f_{2,2} = 0.3576 \\ f_{3,1} = 0.2499 f_{3,2} = 0.2451 \\ f_{4,1} = 0.0537 f_{4,2} = 0.1073 \\ f_{5,1} = 0.2991 f_{5,2} = 0.3389 \\ f_{6,1} = 16.624 f_{6,2} = 10.097 . \end{matrix}

(44)

In accordance with model in (5), only the static and viscous components of the friction torque are taken into account, thus neglecting the nonlinear effects as shown, for example, in Figure 3 for the Coulomb friction. In the identification procedure, static and viscous components will be identified (as shown in (32) and (33)), while the nonlinear effects will be regarded as external disturbances.

Figure 3:

Complete (Stribeck) friction model [22] and Coulomb friction model.

The computed body parameters are given by

\begin{matrix} m_{1} = 21 kg m_{2} = 10 kg \\ c_{1} = {[\begin{bmatrix} - 5 & 0 & 0 \end{bmatrix}]}^{T} kg \cdot m \\ c_{2} = {[\begin{bmatrix} - 2.5 & 0 & 0 \end{bmatrix}]}^{T} kg \cdot m \\ I_{1} = [\begin{bmatrix} 0 & 0 & 0 \\ 0 & 1.25 & 0 \\ 0 & 0 & 13.8167 \end{bmatrix}] kg \cdot m^{2} \\ I_{2} = [\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0.625 & 0 \\ 0 & 0 & 0.8583 \end{bmatrix}] kg \cdot m^{2} . \end{matrix}

(45)

The PD control proposed in [3] has been used to control robot motion in the free space. Excitation trajectories based on fifth order B-splines, as in [23], have been used as reference trajectory for each joint.

Joint angular positions have been sampled at 1 kHz and quantized to simulate a real scenario, by assuming to have a high resolution encoder (1250000 counts/round). Data have been filtered through a Butterworth IIR low-pass filter of fifth-order and cut-off frequency of 20 Hz. Motor and joint angular velocities and accelerations have been computed through a first-order numerical differentiation. Additionally, the effect of sensor noise on the performance of the proposed identification procedure has been evaluated. To this purpose, a Gaussian noise with zero mean value has been added to motor and joint position, velocity, and acceleration and the error percentage between real and estimated dynamic parameters has been measured for five different levels of signal-to-noise ratio (i.e., SN = 20, SN = 70, SN = 80, SN = 100, and SN = 200), in addition to absence of noise. Motion variables have been obtained by means of the robot forward dynamics also accounting for the friction model in (43). In Figure 4 motor torques applied to the simulated manipulator are shown, with the corresponding motion in the joint space. As shown in Figure 5, the maximum difference between link and motor position is 0.39° and 0.10° for first and second joint, respectively.

Figure 4:

Actuation torques for the 2-dof planar manipulator have been generated with the PD control in [3]; corresponding robot motion in the joint space is shown.

Figure 5:

Zoom on the main differences between motor and link positions for the 2-dof planar manipulator.

A total of 8900 measurements have been performed, thus constructing observation matrix W and torque vector T in (36). In order to reject nonlinear friction effects, measurements where joint velocity was lower than a threshold value of 0.5 rad/s have been discarded (a total of 1334 measurements have been discarded). A systematic approach to define this threshold value is presented in [24]. By inspecting matrix W, vector Π^* of 14 identifiable parameters (or linear combination of parameters) has been extracted by means of relation (41), and matrix L has been computed. The obtained values of parameters in Π^* in absence of measurement noise are reported in Table 1, where c_i = [c_{x, i} c_{y, i} c_{z, i}]^T (in SI units).

Table 1:

Results of the identification procedure.

Parameter	Real value	Estimated value

π₁^* = K₁	3000.00 N·m/rad	3000.79 N·m/rad
π₂^* = K₂	1800.00 N·m/rad	1799.79 N·m/rad
π₃^* = m₁ − 4·I_{zz, 1} + 4·I_{zz, 2}	−30.8333 kg	−30.8352 kg
π₄^* = m₂ − 4·I_{zz, 2}	6.5667 kg	6.5670 kg
π₅^* = c_{x, 1} + 2·I_{zz, 1}	22.6333 kg·m	22.6345 kg·m
π₆^* = c_{y, 1}	0.0000 kg·m	− 1.04E^{− 5} kg·m
π₇^* = c_{x, 2} + 2·I_{zz, 2}	−0.7833 kg·m	−0.7835 kg·m
π₈^* = c_{y, 2}	0.0000 kg·m	3.18E^{− 6} kg·m
π₉^* = ϒ₁	21.1800 kg·m²	21.1446 kg·m²
π₁₀^* = ϒ₂	12.1000 kg·m²	12.1036 kg·m²
π₁₁^* = f_{s, 1}	0.3302 N·m	0.1016 N·m
π₁₂^* = f_{s, 2}	0.3576 N·m	0.1249 N·m
π₁₃^* = f_{v, 1}	0.1434 N·m·s/rad	0.1288 N·m·s/rad
π₁₄^* = f_{v, 2}	0.1391 N·m·s/rad	0.1389 N·m·s/rad

As one can easily observe in Table 1, the error between the obtained estimated parameters and the real ones is very low. The normalised mean error is 0.103; it becomes 0.00862 (i.e., 0.862 per cent as percentage error) without the static friction, thus pointing out that probably the neglected nonlinear terms of the adopted model may cause an increase of the error. Furthermore, Table 2 reports the normalised mean error between the estimated parameters and the real ones in the case of noisy measurements of position, velocity, and acceleration, for five different levels of signal-to-noise ratio (i.e., SN = 20, SN = 70, SN = 80, SN = 100, and SN = 200). One can observe that for high values of signal-to-noise ratio, up to SN = 80, the error still remains small and comparable to the absence of noise. For lower values of signal-to-noise ratio, the error becomes very high.

Table 2:

Variation of the estimation error with signal-to-noise ratio.

Signal-to-noise ratio	Normalised mean error

SN = 20	3.84E^{+ 3}
SN = 70	44.50
SN = 80	4.73
SN = 100	0.157
SN = 200	0.111

As aforementioned, the proposed identification procedure of dynamic parameters has also been applied to a 6-dof robot manipulator with elastic joints. It has been modeled as a PUMA 560 with elastic joints. Robot dynamic parameters (except for joint elasticity) are reported in [25]. On the other hand, transmission elasticity for each joint has been assumed to be as K_i = 2000 N·m/rad for i = 1, …, 6.

By analogy with the planar case, excitation trajectories based on fifth-order B-splines have been planned in the joint space and the PD control in [3] has been used to regulate robot position. Motor torques applied to the simulated manipulator and resulting motion in the joint space are reported in Figure 6. For brevity, only the maximum difference between link and motor position for joints 1 and 2 is shown in Figure 7. However, the mean value over the six joints resulted to be 0.09 rad.

Figure 6:

Actuation torques for the 6-dof manipulator have been generated with the PD control in [3]; corresponding robot motion in the joint space is shown.

Figure 7:

Zoom on the main differences between motor and link positions for the 6-dof manipulator.

As for the planar case, observation matrix W and torque vector T in (36) have been constructed. The complete vector of 84 dynamic parameters (i.e., Π) has been estimated through the proposed procedure. The obtained parameters are reported in Tables 3, 4, 5, and 6 together with the values of the real dynamic parameters related to joint elasticity, masses, centers of mass, inertia tensors, motor inertias, and friction. Afterwards, vector Π^* of 62 identifiable parameters (or linear combination of parameters) and matrix L have been extracted. The normalised mean error between estimated and real parameters resulted to be 0.123, that is, very closed to the value obtained in the planar case. The provided simulation tests prove the correctness of the proposed methodology. As future work, the performance on a real robot will be measured.

Table 3:

Real and estimated parameters related to joint elasticity and masses for the 6-dof manipulator.

	π₁	π₂	π₃	π₄	π₅	π₆	π₇	π₈	π₉	π₁₀	π₁₁	π₁₂

Real	2000	2000	2000	2000	2000	2000	0.10	17.4	4.80	0.82	0.34	0.09
Estimated	1999.3	1999.8	1999.6	1999.9	1999.3	2000.1	0.0985	17.41	4.7988	0.819	0.339	0.0889

Table 4:

Real and estimated parameters related to the centers of mass for the 6-dof manipulator.

	π₁₃	π₁₄	π₁₅	π₁₆	π₁₇	π₁₈	π₁₉	π₂₀	π₂₁

Real	2000	0	0	−6.3301	0.1044	3.9585	−0.0974	−0.0677	0.336
Estimated	− 0.25E^{− 4}	− 0.22E^{− 4}	− 0.15E^{− 5}	−6.33087	0.10425	3.95835	−0.09769	−0.0679	0.33587

	π₂₂	π₂₃	π₂₄	π₂₅	π₂₆	π₂₇	π₂₈	π₂₉	π₃₀

Real	0	0.0156	0	0	0	0	0	0	0.0029
Estimated	−0.00015	0.015	− 0.12E^{− 4}	− 0.21E^{− 4}	− 0.15E^{− 4}	− 0.25E^{− 4}	− 0.18E^{− 4}	− 0.25E^{− 4}	0.0026

Table 5:

Real and estimated parameters related to the inertia tensors for the 6-dof manipulator.

	π₃₁	π₃₂	π₃₃	π₃₄	π₃₅	π₃₆	π₃₇	π₃₈	π₃₉	π₄₀	π₄₁	π₄₂

Real	0	0.35	0.2E^{− 4}	0	0	0	1.031	3.727	2.842	0.037	1.440	−0.023
Estimated	− 0.15E^{− 5}	0.34	−0.0015	− 0.1E^{− 4}	− 0.1E^{− 5}	− 0.9E^{− 5}	1.016	3.714	2.838	0.033	1.435	−0.038

	π₄₃	π₄₄	π₄₅	π₄₆	π₄₇	π₄₈	π₄₉	π₅₀	π₅₁	π₅₂	π₅₃	π₅₄

Real	0.090	0.111	0.015	−0.0013	0.0068	0.0047	0.0021	0.0013	0.0021	0	0	0
Estimated	0.077	0.098	0.5E^{− 3}	−0.0026	0.0055	0.0034	0.0008	0	0.0008	−0.0013	−0.0013	−0.0013

	π₅₅	π₅₆	π₅₇	π₅₈	π₅₉	π₆₀	π₆₁	π₆₂	π₆₃	π₆₄	π₆₅	π₆₆

Real	0.3E^{− 4}	0.4E^{− 4}	0.3E^{− 4}	0	0	0	0.0002	0.0002	4E^{− 05}	0	0	0
Estimated	0.1E^{− 4}	0.9E^{− 4}	0.9E^{− 4}	− 0.1E^{− 4}	− 0.1E^{− 4}	− 0.1E^{− 5}	−0.0011	−0.0011	0.12E^{− 4}	0.13E^{− 4}	− 0.13E^{− 4}	− 0.11E^{− 4}

Table 6:

Real and estimated parameters related to motor inertias and friction for the 6-dof manipulator.

	π₆₇	π₆₈	π₆₉	π₇₀	π₇₁	π₇₂	π₇₃	π₇₄	π₇₅

Real	0.0002	0.0002	0.0002	3.3E^{− 5}	3.3E^{− 5}	3.0E^{− 5}	0.395	0.126	0.132
Estimated	0.0252	0.0252	0.0252	0.25E^{− 4}	0.25E^{− 4}	0.15E^{− 4}	0.396	0.127	0.147

	π₇₆	π₇₇	π₇₈	π₇₉	π₈₀	π₈₁	π₈₂	π₈₃	π₈₄

Real	0.0112	0.0093	0.0039	0.0014	0.0008	0.0014	7.12E^{− 5}	8.26E^{− 5}	3.67E^{− 5}
Estimated	0.0118	0.0140	0.0040	0.0006	0.0005	0.0017	2.01E^{− 5}	5.95E^{− 5}	1.42E^{− 5}

7. Conclusions

A novel procedure for identifying dynamic parameters for robots with elastic joints has been proposed. The proposed procedure is based on the Lagrangian formulation of the dynamic model of the manipulator, accounting for robot elastic energy, in addition to kinetic and gravitational energy contributions. This feature represents the actual main novelty of this work, since no systematic approaches for parameter identification have been reported in the literature for robots with elastic joints.

Each term of the dynamic model has been analyzed in order to linearize it and express the vector of external and motor torques as the product of a regressor matrix by a vector of dynamic parameters to identify. Special attention has been paid to the extraction of the regressor matrix, in order to facilitate parametric identification. Also a method is proposed to identify the category (unidentifiable, linearly dependent, or else independent) to which a parameter belongs. Two robotic structures with elastic joints of different complexity have been simulated in order to validate the procedure of parametric identification, that is, a 2-dof planar manipulator and a 6-dof manipulator. The PD control in [3] has been used to move the robot in the free space; the simulation of the forward dynamics of the manipulator has permitted to collect joint and motor positions, velocities, and accelerations during motion and consequently apply the identification procedure with and without measurement noise.

The obtained results have shown that the proposed procedure leads to a correct identification of the manipulator dynamic parameters with a very low error; the mean normalised error between actual parameters and estimated ones is 0.103 for the planar manipulator and 0.123 for the 6-dof manipulator. The percentage error decreases to a value around 0.862 percent if the static friction is excluded (which is more affected from the neglected nonlinearities of the adopted friction model). In presence of measurement noise, the error is still low for signal-to-noise ratios higher than 80. The correctness of the analytical formulation of the regressor matrix and, consequently, of the parametric identification procedure is thus assessed and the application on a real robot with elastic joints can be envisaged as future activity.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Appendices

Authors' Contribution

Loredana Zollo and Edoardo Lopez contributed equally to this paper.

1

In (18) and operator (a)[⊳ ⊲] has been introduced, which generates a matrix (3 × 6) from a (3 × 1) vector a. It is defined as

(*)

{(a)}_{[⊳ ⊲]} = [\begin{bmatrix} a_{1} & 0 & 0 & a_{2} & a_{3} & 0 \\ 0 & a_{2} & 0 & a_{1} & 0 & a_{3} \\ 0 & 0 & a_{3} & 0 & a_{1} & a_{2} \end{bmatrix}] .

Given a (3 × 3) symmetric matrix S of elements s_ij, operator (a)_{[⊳ ⊲]} allows expressing product Sa as

(**)

S a = {(a)}_{[⊳ ⊲]} {[s_{11} s_{22} s_{33} s_{12} s_{13} s_{23}]}^{T} .

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Identification of Dynamic Parameters for Robots with Elastic Joints

Abstract

1. Introduction

2. Notation

3. Dynamic Model of Robot Manipulators with Elastic Joints

4. Linearization of Robot Dynamic Model

5. Parametric Identification Procedure

6. Validation and Results

7. Conclusions

Conflict of Interests

Footnotes

Appendices

Authors' Contribution

1

References