2.2.1 Impedance model in Cartesian space

A desired impedance equation for a robot manipulator in Cartesian space is designed by

(2.1)

where x∈ ℜ⁶,∈ ℜ⁶ and ∈ ℜ⁶ are the position, velocity and acceleration vectors, respectively. M_x (θ) ∈ ℜ^6 × 6 is the inertia matrix, θ∈ ℜ⁶ is the vector of generalized joint coordinates describing the pose of the manipulator. F∈ ℜ⁶ is the force-moment vector acting on the endeffector defined by F^T = [f^T n^T], where f∈ ℜ³ and n ∈ ℜ³ are the force and moment vectors, respectively. K_f = diag(K_f1,…,K_f6) is the force feedback gain matrix. x_d, and F_d^T = [f_d^Tn_d^T] are the desired position, velocity, force and moment vectors; B_d = diag(B_d1,…,B_d6) and K_d = diag(K_d1,…,K_d6) are the coefficient matrices of desired damping and desired stiffness, respectively. S = diag(S₁,…, S₆) and I are the switch matrix and identity matrix. It is assumed that K_f, B_d, and K_d are all positive-definite diagonal matrices. Note that if S = I, then Eq. (2.1) becomes a compliance control system in all directions; on the other hand if S is the zero matrix, it becomes a force control system in all directions. It is assumed that the acceleration of Eq. (2.1) is very small, so the inertia term can be ignored. Defining X:=x − x_d, then Eq. (2.1) can be rewritten as

(2.2)

where

(2.3)

In general, Eq. (2.2) is solved as

(2.4)

In the following, we consider the form in discrete-time system using a sampling width ∆t. It is assumed that u and A are constant at ∆t(k–1) ≤ t < ∆tk. Defining X(k) = X(t)|_t = ∆tk(k = 0,1,…), Eq. (2.4) is transformed into the discrete-time domain description:

(2.5)

Remembering X(k) = x(k) − x_d(k) leads to

(2.6)

where x(k) consists of position vector [x(k) y(k) z(k)]^T and orientation vector [ϕ(k) ξ(k) ψ(k)]^T expressed by Z-Y-Z Euler angles. In order to realize the impedance given by Eq. (2.1), x(k) is given to the reference of robotic servo system every sampling period.

2.2.2 How to generate the reference for servo controller

To simulate the impedance model following force control method using an industrial robot model, the manipulated variable x(k) calculated from Eq. (2.6) has to be transformed into joint driving torques. Accordingly, we propose a transformation technique where x(k) is given to the reference value of the Cartesian-based servo controller. The block diagram of this transformation is illustrated in Fig. 2.1. The joint driving torques transformed from x(k) control each joint independently.

First of all, a velocity and acceleration are generated by using a discrete time k and sampling width Δt.

(2.7)

(2.8)

For example, in the case that the resolved acceleration control law is used in the servo system of an industrial robot, x(k), (k) and (k) generated from Eqs. (2.6), (2.7) and (2.8) are respectively given to the references x_r, and of the servo system, so that the joint torques are generated from

(2.9)

where, M(θ) ∈ ℜ^6 × 6 is the inertia term in joint space; J^−l(θ) is the inverse matrix of Jacobian with the relation = J(θ) . K_v = diag(K_vl,…, K_v6) and K_p = diag(K_p1,…, K_p6) are the gains of velocity and position, respectively. and G(θ) are the Coriolis and centrifugal forces term and gravity term in joint space, respectively. Note that θ, , x and in Eq. (2.9) are actual values, i.e., controlled variables. The non-linear compensation terms H (θ, ) and G(θ) are effective to achieve a stable control system.

Thus, the proposed method allows us not only to implement the impedance model following force controller in the computer simulations, but also to easily apply it to industrial robots with an open-architecture control system.

2.2.3 Design of desired damping considering the dynamics of environment

In the previous section, we have introduced the impedance model following force control for articulated industrial robots with an open-architecture controller. The problem now is how to tune the desired damping that mainly affects the force control performance. In this section we propose a tuning method of desired damping using known physical parameters. Fig. 2.2 shows an example where an industrial robot is making contact with an object inclined at q degrees from the floor. It is assumed that a stiff end-effector is fixed to the tip of the robot arm as shown in Fig. 2.3. The contact force from the normal direction of the slope is controlled to converge to a reference value ^Sf_d. Here, the left superscript S means the value in sensor coordinate system. The end-effector is controlled to exert a small force on the object along the normal direction.

The rotation matrix in the sensor coordinate system based on the base coordinate system is described as

(2.10)

Therefore, the desired force vector in the base coordinate system is obtained by

(2.11)

For example, is transformed into . Here, it is assumed that the contact force f = [f_x f_y f_z]^T acting on the end-effector is represented by

(2.12)

where B_m = diag(B_m1, B_m2, B_m3) and K_m = diag(K_m1, K_m2, K_m3) are the object’s viscosity and stiffness coefficients to be positive-definite diagonal matrices. It is also assumed that the tip of the robot arm contacts the object at the position x_m^T = [x_m y_m z_m]. The transition from unconstrained motion to constrained motion is simply considered by Eq. (2.12), since it has been reported that analyzing the contact force at impact with proper models is difficult and complex [17].

In force control mode, the desired damping considering the critical damping condition of Eq. (2.1) is given by

(2.13)

Here, i(i = 1,2,3) denotes the i-th diagonal element of each matrix. When the impedance model following force controller is applied, is given as a reference of desired damping so that promising force control suppressing overshoots and oscillations can be realized. It should be noted that when this tuning method is used, physical parameters of environments, i.e. B_mi and K_mi, are indispensable to calculate .

2.4.1 Design of fuzzy environment model

The fuzzy environment model (FEM), based on fuzzy reasoning theory, estimates the stiffness of environment. We design the FEM by using the simple fuzzy reasoning theory, which is regarded as a kind of model, developed by Takagi et al. [21]. In this section, we describe how to construct the FEM giving an example of force control shown in Fig. 2.2. According to the former definition, the position vector and the force vector at the tip of the robot arm are given with x^T = [x(t) y(t) z(t)] and f^T = [f_x(t) 0 f_z(t)], respectively. The motivation behind the proposed concept is to simplify the construction of the FEM, so each reasoning in the x-or z-direction is conducted independently. Fuzzy inputs for reasoning are defined by

(2.15)

(2.16)

If the information on only x-direction is used, the FEMs are described by

Rule 1:

IF x*(t) is , THEN + K_m¹ (x − x_m) = − f

Rule 2:

IF x*(t) is , THEN + K_m² (x − x_m) = − f

Rule L:

IF x*(t) is , THEN + K_m^L (x − x_m) = − f

Where (i = 1,…,L) is the antecedent fuzzy set and L is the number of labels. If the x-directional element K_m1 of stiffness coefficient matrix K_m is estimated in parallel with above fuzzy environment models, the reasoning rules are represented by

Rule 1:

IF x*(t) is , THEN K_m1 = K_m11

Rule 2:

IF x*(t) is , THEN K_m1 = K_m12

Rule L:

IF x*(t) is , THEN K_m1 = K_m1L

where K_m1i(i = 1,…,L) denotes the consequent constant. Using these rules, the estimated stiffness in the x-direction is computed by the weighted mean method expressed as

(2.17)

where p_i {x*(t)} denotes the normalized confidence given by

(2.18)

with the antecedent confidence {x^∗(t)}described by the following Gaussian membership function

(2.19)

where, α_j is the center of membership function and β_j is the reciprocal value of standard deviation, and j(j = 1,…,L) denotes the number of fuzzy rules. The Gaussian type membership function is shaped by only the center of membership function and the reciprocal value of standard deviation. In optimizing the FEM using genetic algorithms (GA), the confidence is calculated after the genotype of each FEM is decoded to the phenotype. In this case, the Gaussian type membership function allows the system to easily calculate the confidence, compared with triangular type membership functions. Because of this advantage, we use the Gaussian type membership function. The weighted mean method is generally used in the simple fuzzy reasoning method. This method requires no centroidal calculations, so that high speed operation and analysis can be realized, compared to the min-max centroidal method or the product-sum cen- troidal method. In the same manner, the estimated stiffness in the z-direction is computed by giving Eq. (2.16) to the input of fuzzy reasoning. Fig. 2.5 shows the block diagram of the impedance model following force controller with the FEM. The FEM estimates the environmental stiffness sensing the force measurements and provides the desired damping based on Eq. (2.13) to maintain the system in critically damped condition. The impedance model following force controller generates the manipulated variable x(k) for the servo system with .

As can be seen from the above arguments, the key point for the design of the FEM is to appropriately adjust α, β in the antecedent part and consequent constant given by γ. However, the effective design method for such parameters has not been developed. This means that if we want to use the FEM then the complicated parameter tuning through many simulations or experiments by trial and error is unavoidable. In the remainder of this paper, we propose a systematic tuning method for the FEM using a GA approach.

2.4.2 Offline learning of FEMs using genetic algorithms

Fortunately, if the model of a manipulator used is known, then the robust FEM can be derived in advance using the GA. Before optimizing the FEM by using GA, we design the base of a FEM as follows: the type of membership functions is Gaussian; the fuzzy rule number is five; consequent parts have constant values; and so on. The GA is used to automatically adjust the parameters associated with the fuzzy rules. In other words, the GA optimizes the parameters of fuzzy rules such as the center of membership function α_j, reciprocal value of standard deviation β_j and constant value of consequent part γ_j.α_j, β_j and γ_j are the phenotypes of FEM, and each phenotype is encoded into the genotype of 16 bits binary code. An individual, i.e. a FEM, is composed of 5 fuzzy rules (L = 5) as shown in Fig. 2.6.

In our proposed FEM, a parameter space model deals with the viscosity and the stiffness independently. It is also assumed that the parameter space can be partitioned abstractly and formally, in which the combinations of the viscosity and the stiffness are not considered. In this case, the partition number of the parameter space is equal to the number of fuzzy rules. We empirically designed the FEM (i.e., an individual) with five fuzzy rules. For example, we can design the fuzzy labels as Further Soft, Rather Soft, Approximately Same, Rather Stiff, and Further Stiff. Of course, it is possible to use more fuzzy rules, e.g. 7 fuzzy rules for finer reasoning. Each individual is evaluated by applying the proposed method shown in Fig. 2.5 to the force control problem given in Fig. 2.2. The evaluation function E_v of each individual is the sum of squared error between the force facting on the tip of the robot arm and the reference value f_d at every sampling time ∆tk, which is calculated by

(2.20)

where T is the total simulation time. This is a minimization problem and therefore if the individual has the minimum E_v then it can survive as an elite. When an overshoot, oscillation or a non-contact state appears, a large dummy fitness (penalty) is given to the individual to be excluded, because such an individual is undesirable in a stable force control system. The parameters used in the GA operation are tabulated in Table 2.3. To learn the FEM, the seven kinds of environments shown in Table 2.4 are selected. Because of the aforementioned reason, the viscosity of the environments was fixed to 15 Ns/m. In addition, it was assumed that the stiffness and the viscosity of the environments have the same characteristics in x- and z-directions, respectively.

Here, we examine the computational cost for learning. When a PC (CPU: Intel Celeron 1.73 GHz) is used on Windows XP Professional, it takes about 15 seconds to evaluate one individual, i.e., to integrate the dynamic model given by Eq. (2.14) for the simulation time of 2 seconds by using a Runge-Kutta algorithm. Hence, at least 15 seconds × 60 individuals × 100 generations × 7 environments = 175 hours are required to finish the whole learning.

2.4.3 Learning results

Figure 2.7 shows the relationship between the true stiffness K_m1 (or K_m3) of an environment and the estimated stiffness (or ) in a steady state which has been obtained with best individuals. Figure 2.8 shows the relationship between the K_m1 (or K_m3) and the desired damping (or ) in the x- or z-direction given from Eq. (2.13). In Fig. 2.8, the solid lines show the desired damping (:, :) in using and . On the other hand, the dotted lines represent the one (∆: , o: ) in using K_m1 and K_m3 in which it is assumed that K_m1 and K_m3 are known. It can be observed from the result that they intersect at near the stiffness 1000 N/m. These learned results suggest that the desired damping, considering the critical damping condition with K_m1 (or K_m3), could achieve stable force control without overshoots and oscillations if the K_m1 (or K_m3) is under 1000 N/m and its neighborhood. However, if the K_m1 (or K_m3) is larger than the range, then a larger desired damping is needed to suppress the overshoots and oscillations. This means that if the K_m1 (or K_m3) becomes larger than 1000 N/m and its neighborhood, then the environmental stiffness in Eq. (2.13) must be estimated to be much larger than the true one in order to realize a desirable response. Furthermore, it can also be observed that there is a proportional relation between the real stiffness and the desired damping coefficient. It is supposed that the slope would change depending on how to evaluate and select the individuals.

As an example of the learning effectiveness on seven kinds of environments, the average and minimum values of fitness are shown in Fig. 2.9, which is the evolutionary history in the case of K_m1 = K_m3 = 17500 N/m. It is recognized that more excellent individuals have been selected through the alternation of generations with GA operations. Figure 2.10 and Table 2.5 respectively show the antecedent membership functions and consequent constants which mean the best elite individual (a best FEM) after 100 generations. Figure 2.11 shows the simulation results using the best individual in the case of K_m1 = K_m3 = 17500 N/m. Figure 2.11(a) shows the force control result using the desired damping which is calculated by substituting the estimated stiffness into Eq. (2.13). Certainly, the desirable performance without overshoots and oscillations is observed. At this time, , and , are varied as shown in Figs. 2.11(b), (c), respectively.

It should be noted, however, that there are nonoverlapping membership functions in Fig. 2.10. In this environment, since fuzzy inputs are generated around 17500 N/m, the non-overlapping part has no impact on the response. However, the robot must deal with all supposed cases (i.e., range) in real applications, so that some flexibility (referred to as generalization) is required. At this stage, seven FEMs exist as many as the learned environments. Of course, it is not guaranteed that the desired performance can be achieved except for the learned environments. The most important function required for the FEMs is that the desired damping can be generated for all environments with any stiffness within an assumed range. In order to address this function, we consider on how to construct the learned FEMs for the generalization, where the antecedent part has no non-overlapping membership function.

2.4.4 Generalization of FEMs

Each FEM discussed in the previous section has five fuzzy rules. If those are simply combined, then the number of fuzzy rules becomes 35 and the antecedent membership functions overlaps each other on the support set. When the combined FEMs are used, it might be that a suitable force control cannot be achieved even under a learned environment. The reason is that the fuzzy rules undesirably influence each other. Therefore, we propose a design method for the generalization which can make the FEMs work well for any environment only if the stiffness is within the range (e.g., from 3500 N/m to 21000 N/m) shown in Fig. 2.7. In order to generalize the FEMs learned under seven fixed environments, it should be considered how to interpolate the antecedent parts between two adjacent FEMs. The approach for the generalization is as follows: Although the number of learned environments is seven as shown in Table 2.4, the generalized fuzzy environment model (GFEM) is composed of six results except for Table 2.4(1) (i.e., K_m1 = K_m3 = 350). It is assumed that Table 2.4(1) is included in Table 2.4(2). The stiffness values of Table 2.4(2)-(7) are used for the centers of antecedent membership functions in the GFEM, and also a suitable value is given to the standard deviation of Gaussian membership functions so that there is no nonoverlapping area. On the other hand, in the design of the consequent part, the estimated stiffness coefficients shown in Fig. 2.7 are directly used for the consequent constant values.

The concrete design method is that the antecedent membership functions are placed within the range (e.g., from 3500 N/m to 21000 N/m) with the same interval as shown in Fig. 2.12, and the estimated values of environmental stiffness shown in Fig. 2.7 are set to the corresponding consequent constants. Taking account of the fact that the number of FEMs (from 3500 N/m to 21000 N/m) integrated for the generalization is six, we design the GFEM with six fuzzy rules. K_m1 (or K_m3) shown in Table 2.4(2)-(7) are used as the center of each membership function so that they are allocated on the support set [3500,21000] with an equal interval. Furthermore, we give 1750 to the standard deviation in order that the membership functions disposed side by side can intersect at the fitness 0.5 each other. In the design of the consequent part, the consequent models as many as the antecedent labels are prepared. The estimated stiffness coefficients, which are obtained by giving the center values of the antecedent membership functions to the horizontal values shown in Fig. 2.7, are used for the consequent constants as tabulated in Table 2.6. Fig. 2.12 and Table 2.6 show the designed antecedent membership functions and consequent constant values, respectively. The designed GFEM requires no additional acquirements and modifications of fuzzy rules during the execution of force control.

2.4.5 Results of hybrid position/force control

In this section, the property of robustness is proved using the impedance model following force controller with the GFEM. Here, a simple hybrid position/force control problem shown in Fig. 2.13 is conducted under the four kinds of unlearned environments shown in Table 2.7. The degree of slope is set to 30°. The contact force in the z-direction and the trajectory from point P to point Q in the y-direction in the sensor coordinate system are simultaneously controlled. To perform this task, the switch matrix S is set to diag(0, 1, 0). The desired contact force is given as

(2.21)

Using Eq. (2.11), ^Sf_d^T(t) is transformed to f_d^T(t) in the base coordinate system such as

(2.22)

Where Z-Y-Z Euler angles, which represent the orientation of the tip of the robot arm, are fixed to [0 150° 0].

The experimental results of hybrid position/force control are shown in Figs. 2.14 and 2.15. It was confirmed that the contact forces could follow the changes of the reference without overshoots and oscillations under four environments. In this case, the desired damping coefficients in the x- and z-directions are generated from the GFEM as shown in Figs. 2.16 and 2.17, respectively. Figure 2.18 demonstrates another result, in which the desired contact force was set to larger magnitudes. From these simulation results, it was proved that the proposed GFEM could systematically generate the desired damping for stable force control even under unlearned environments.

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