Parameter Identification

This section presents empirical models for the friction, stiffness of the manipu-lator joints and the springs for gravity compensation of the CROPS manipumanipu-lator prototype 2. Kinematic and dynamic parameters are given in appendix A.

Friction

Revolute Joints For the large and medium sized drive modules (joint 2-6) of the CROPS manipulator 2, the following friction model for revolute joints is pro-posed by Baur, Dendorfer, et al. (2014):

T_f,i,Baur=−^sgn(q˙_inc,i)(T^¯_f,i,0+µ¯|^Tl,i|)−^b¯q˙inc,i−^Tf,S−^T¯f,i,0

1+ (^q˙inc,i_q˙S )^{2 (3.12)}

Joint T¯_{f i,0}[Nm] µ¯[−] b^¯[^Nms_rad ] T_f,S[Nm] q˙_S[^rad_s ] 2 2.76·^{10−1 2.10}·^{10−3 5.24}·^{10−4 1.68}·^{10−1 5.33}·¹⁰¹ 3-4 2.79·^{10−1 3.75}·^{10−4 4.68}·^{10−4 1.28}·^{10−1 8.17}·¹⁰¹ 5 1.04·^{10−1 5.20}·^{10−3 1.42}·^{10−4 6.03}·^{10−2 5.45}·¹⁰¹ 6 1.36·^{10−1 4.40}·^{10−3 1.95}·^{10−5 4.77}·^{10−2 6.52}·¹⁰¹

Table 3.4: Parameters for the friction model eq. (3.13) according to Baur (2015, p.77).

Thus, the friction torque depends on the joint velocity on the motor shaft ˙q_inc,iand the external loadT_l,i. The parameters[T^¯_f,0, ¯µ, ¯b,T_f,S, ˙q_S] were experimentally de-termined on a drive module testbed (cf. table 3.4). However, this model assumes the motion direction to be non-negative, i.e. ˙q ≥0. This is corrected for arbitrary directions of ˙q by introducing the coefficient sgn(q˙_inc,i). Furthermore, it is ad-justed by a piecewise defined correction termχ_i for the assembled manipulator.

Hence, the friction law results in:

T_f,i =−^sgn(q˙_inc,i)(T^¯_f,i,0+µ¯|^Tl,i|)−^b¯q˙inc,i− sgn(q˙_inc,i)^Tf,S−^T¯f,i,0

1+ (^q˙inc,i_q˙S )² −^χi for joint 2 – 6. (3.13)

For the small drive modules on the manipulator’s wrist (joint 7-9), a model based on the gear manufacturer’s catalog was developed by Buschmann (2010). Param-eters[T_{f i,0},µ,bg,γ_f]of the original model were identified for the CROPS manip-ulator by Baur (2015) (cf. table 3.5). Furthermore, the correction term χ_i for the assembled manipulator is added as well:

T_f,i =−^sgn(q˙_inc,i)(T_{f i,0}+µ|^Tl,i|)

−(b_g+γ_f|^Tl,i|)q˙_inc,i−^χi for joint 7 – 9 (3.14) The piecewise defined correction termχ_iwith its parametersχ_i,(·)(cf. table 3.6) is defined by

χ_i =

(χ_{i, ˙qi<0}·^tanh(N_i·^q˙i) if ˙q_i <0

χ_{i, ˙qi>0}·^tanh(N_i·^q˙i) else. (3.15)

Joint T_f,i,0[Nm] µ[−] bg[^Nms_rad ] γ_f[_rad^s ] 7-9 1.88·^{10−2 1.04}·^{10−4 2.85}·^{10−5 1.50}·¹⁰⁻⁵

Table 3.5:Parameters for the friction model eq. (3.14) according to Baur (2015, p. 125)

Joint 2 3 4 5 6 7 8 9 χ_{i, ˙qi<0} [Nm] +1.00 +7.00 +11.0 −^5.00 −^2.00 +1.75 +0.60 +0.70 χ_{i, ˙qi<0} [Nm] +1.25 +7.00 +11.0 −^4.50 −^2.50 +1.50 +1.00 +0.80 Table 3.6: Correction Term of the friction model for the assembled manipulator.

Prismatic Joint The CROPS manipulator 2 uses the linear axis FESTO EGC-TB for its prismatic joint. Using the model presented by Baur (2015), the friction force F_f is calculated by

F_f =−^sgn(q˙₁)F_f,0−^bq˙1 (3.16)

with the Coulomb friction constant F_f,0 and the viscous friction gain b. Baur (2015) identified the parameters of the prismatic joint oriented horizontally with-out any force. This setup does not correlate to its normal operation. Therefore, these parameters are recalculated by setting the joint vertically and adding a con-stant mass (with a comparable weight to the manipulator arm) to joint 1. As depicted in fig. 3.8 the orientation shows a major influence on the friction param-eters (cf. table 3.7).

Gravitational Force Compensation

In order to compensate for the gravitational load of the arm, eight constant force springs are attached to the prismatic joint of the CROPS manipulator 2. Two of these springs were tested individually by measuring the force while the joint 1 moves up and down. Fig. 3.9 shows typical results for two measurements. Both springs show a similar behavior, due to Coulomb friction. Their respective forces F_sp depend on the direction of the movement. Furthermore, the right spring shows an abnormality, which is explained by a buckling located in the spring.

The average force for all measurements is−50.8 N per spring.

Joint Stiffness

In practice, each joint has a limited stiffness. Hence, the difference ∆q_i between the joint position on the output shaft q_i,abs and on the motor shaft q_i,inc gives information about the total acting torqueT_i. While the positions on the motor and output shaft can be measured, a model for the stiffness ˜c_i has to be determined.

This section introduces an empirical model for the joint stiffness ˜c_i. By analogy to a torsional spring, the quantities can be written as

T_i =c˜_i ∆ϕ_i

∆ϕ_i :=qabs−^qi,inc. (3.17)

−^0.6 −^0.4 −^{0.2 0 0.2 0.4 0.6}

−²⁰⁰

−¹⁰⁰ 0 100 200

q˙1[^m/s]

Ff[N] Measurement (Vertical)

Parameter Baur Parameter Schütz

Figure 3.8:Friction parameters for the prismatic joint of the CROPS manipulator 2

Parameter F_f,0 [N] b [^Ns/m] Horizontal Setup (Baur) 67.41 22.97 Vertical Setup (Schütz) 119.1 85.01 Table 3.7:Friction parameter sets for prismatic joint.

LeftSpring

0 20 40 60 80 100

−⁶⁰

−⁵⁰

−⁴⁰

Time[s]

Force[N]

−^0.4

−^0.2 0 0.2

Position[m]

RightSpring

0 20 40 60 80 100

Time[s] Measured Force Position Average Force

Figure 3.9: Forces of two gravitation compensation springs attached to the prismatic joint of the CROPS manipulator 2. The force Fsp is measured (gray) while the joint 1 moves up and down. The position is indicated in green. The average force of the measurement is given in blue. Both springs are assumed to show a similar behavior.

The right spring shows an abnormality (purple) at one position due to a buckling in the spring.

−^{5 0 5}

·¹⁰⁻⁴

−²⁰ 0 20

∆ϕ₂[rad] AppliedTorqueTi,app[Nm]

Joint 2

−^{5 0 5}

·¹⁰⁻⁴

−⁵ 0 5

∆ϕ₆[rad] Joint 6

HD Gear Stiffness Measurement Linear Fitting

Figure 3.10:Comparison of measured stiffness (gray) from quasi-static experiment and HarmonicDrive catalog data (green). Stiffness measurements for joint 2 (large) and 6 (medium size). Maximal load in the measurements corresponds to40%of the nominal torque of joint 2 and30%of joint 6.

In the following, ˜c_i is identified in quasi-static experiments. Known forces are applied to the manipulator, which can be mapped to the single joints as general-ized torques Q_i,app. Fig. 3.10 shows the results for a large drive module (joint 2) with a gear reduction of N2 = 50 and for a medium size drive module (joint 6) with N6 = 100. In this context, the tangent stiffness ˜c_i is defined as the gradient

∂Ti,app

∂∆ϕ_i . A linear fitc_i of the measurement data is shown which is considered to be an appropriate approximation for ˜c_i, i.e. ˜c ≈ c. Stiffness values for the gears given by the manufacturer are plotted for comparison. The measurements show a hysteresis, which is larger in joint 2 than in joint 6. Experimental data for joint 2-8 are given in appendix B. Results varied among the different measurements, their boundary values[c¯_i,min, ¯c_i,max]as well as the total average valuec_iof all mea-surements are listed in table 3.8. Additionally, the stiffness values of the gears’

manufacturer are specified, as well as the boundaries for the piecewise definition for low, medium and high loads. Note that the measured stiffness of the large modules coincides with the middle to high range of the catalog stiffnesses. The measured stiffness of medium sized modules is significantly lower compared to the catalog data.