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Four relevant inertias can be distinguished in the setup. These inertias are: Jm+e = 7·10−6_kgm2 _{for the motor and encoder combination, J}

pulley = 4.6·10−7kgm2 for the two half pulleys, Jos = 2.7·10−4kgm2 for the outer segment part and Jis+ee = 4.4·10−4kgm2 for the inner segment part with end effector. The inertia of all components together is Jtot = 1.4 · 10−3 kgm2, equal to an apparent mass of m = 0.25 kg at the tip of the end effector. This value can be reduced to 0.21 kg (1.2 · 10−3 _kgm2_{) by using an aluminum} end effector, although it would result in a statically unbalanced system and is therefore not done.

The relevant torsional stiffnesses are kms= 1.5 · 102 Nm/rad for the motor shaft, kdyn = 2.9·103

Nm/rad for the Dyneema®_{wire and k}

ts = 3.5·103Nm/rad for the torque sensor. Furthermore, there is a deflection of the motor shaft under load. The bending stiffness is cms,b = 1.4 · 106 N/m. This equals a torsional stiffness of kms,b = 7.9 · 103 Nm/rad. The deflection of the main shaft and the bearing stiffness are left out of consideration

as these stiffnesses are an order higher.

Figure 4.10 represents a simplified dynamic model of the setup. The bending stiffness of the motor shaft is placed in series with the stiffness of the Dyneema® _{wire. The}

following set of equations describes this model: Jm+e i ·ϕ¨m+e = U i − kms i (ϕm+e−ϕpulley) (4.5) Jpulley i ·ϕ¨pulley = kms

i (ϕm+e−ϕpulley) − kcapstan(i · ϕpulley−ϕos) (4.6) Jos·ϕ¨os = kcapstan(i · ϕpulley−ϕos) − kts(ϕos−ϕis+ee) (4.7)

Jis+ee·ϕ¨is+ee = kts(ϕos−ϕis+ee) (4.8)

with: kcapstan =  1 kms,b + 1 kdyn −1 Jm+e Jpulley Jos Jis+ee kms kms,b kdyn kts U ϕm+e ϕpulley ϕos ϕis+ee i = 1 10 Jm+e = 7· 10−6 kgm2 kms= 1.5 · 102 Nm/rad Jpulley = 4.6 · 10−7 kgm2 kms,b= 7.9 · 103Nm/rad Jos= 2.7 · 10−4kgm2 kdyn= 2.9 · 103Nm/rad Jis+ee = 4.4 · 10−4 kgm2 kts= 3.5 · 103Nm/rad

Figure 4.10 /Dynamic model of the 1 DoF setup. The input of the actuator and the rotation of the end effector are indicated by respectively U and ϕ. The capstan drive has a ratio of i = 1/10.

Equations (4.5-4.8) are used to draw the Bode plot in figure 4.11. The figure includes the frequency response function (FRF) from motor to encoder and the FRF from motor to end effector for a frequency up to 2 kHz. Damping is left out of consideration. Eigenfrequencies are visible at 320 Hz and 810 Hz. The third eigenfrequency at 3.2 kHz is not drawn.

The FRF from motor to encoder is also measured1 _{to verify the dynamical model, see}

100 101 102 103 −150 −100 −50 0 50 frequency [Hz] Magnitude [dB] H_{motor → encoder}

H_{motor → end effector}

Figure 4.11 /Frequency response of the dynamic model of the 1 DoF rotational setup for a frequency up to 2 kHz. Eigenfrequencies are visible at 320 Hz and 810 Hz.

figure 4.12. Eigenfrequencies are visible around 310 Hz and 820 Hz. The behavior is typically a result of the collocated sensor-actuator location. The small peak at 150 Hz has to do with the fact that the Bode plot is a combination of two individual measurements, one from 1-150 Hz and one from 150-2000 Hz.

100 101 102 103 −200 −150 −100 −50 0 frequency [Hz] Magnitude [dB] 100 101 102 103 −150 −100 −50 0 50 100 150 frequency [Hz] Phase [Degrees]

Figure 4.12 / Plant FRF of the master and slave device. Eigenfrequencies are visible at 310 Hz and 820 Hz.

The quality of the haptic feedback is also influenced by the device friction. The friction force at the tip of the end effector is approximately 0.65 N for a clockwise (CW) segment

rotation (end effector moves downwards) and 0.45 N for a counterclockwise (CCW) segment rotation (end effector moves upwards) [9]. The friction torque of the DC motor itself is respectively 3.5-4 mNm and 2-2.5 mNm. This equals a force of approximately 0.5 N (CW) and 0.3 N (CCW) at the tip. The friction of the setup itself, as caused by the capstan drive and the two angular contact bearings, hence is in the order of 0.15 N. An explanation for the difference in friction torque can be found in the placement of the brushes with respect to the collector segments. According to [55], the two brushes are made as lever brushes. This means that each brush is supported by a hinge. A spring presses the brush against the collector segments. If the rotation point of the brush is not placed on a line tangential with these segments, then the normal force and friction force depend on the direction of rotation of the rotor. With a continuous force level of 10 N, the friction level does not fulfill the ≤1% requirement as stated in the design specifications in section 4.2.

Another aspect is the torque ripple of the motor. The theoretical ripple of the motor depends on the number of commutation points. For the motor applied, the theoretical ripple is only 0.75% [54]. In practice, the ripple is increased by for instance brush and commutator arcing. The fact that the user can feel these disturbances makes that a DC motor with graphite brushes is not a good choice for a haptic setup.