3.1. Model description

When the robot moves in the horizontal plane with motion law $x_b(t)$ , the suspended orthosis behaves like a pendulum. To describe the transverse vibrations, the planar model with two cables of Fig. 5 was developed. The planar model is composed of two upper cables ( $L_{30}$ and $L_{40}$ ) which connect the pulleys to the magnetic hooks (mass $m_3$ and $m_4$ ). The negligible influence of the pulleys makes it possible to neglect the effect of the horizontal cables $L_1$ and $L_2$ that connect the motors to the pulleys, since they do not influence transverse vibrations. A payload equivalent to 2/3 of the expected load on the orthosis (mass $M$ ) is supported by two lower cables with length $L_{50}$ and $L_{60}$ . Similarly, the values of $k_x, k_y$ and $c_x, c_y$ adopted in the simulations are 2/3 of the values in Table II.

In the present analysis, the system is considered symmetric with $L_{30}=L_{40}$ , $L_{50}=L_{60}$ and $m_3=m_4$ . In the reference configuration, the rotations of the payload and the cables are equal to $\varphi _2 = 0$ and $\varphi _3 = \varphi _4 = \frac{3}{2}\pi$ . The quadrilateral $OABC$ is a parallelogram. The vibrations of the system about the reference configuration are described by the three coordinates depicted in Fig. 5 ( $\theta _3$ , $\alpha _3$ and $\alpha _4$ ). $\alpha _3$ and $\alpha _4$ are the rotations of the upper cables due to the lateral motion of the hooks. Since small oscillations are considered, these simple relations between the rotations of the upper and lower cables hold:

(1) \begin{equation} L_{30}\alpha _3 = L_{50}\alpha _5 \end{equation}

(2) \begin{equation} L_{40}\alpha _4 = L_{60}\alpha _6 \end{equation}

$\theta _3$ is the rotation of the quadrilateral $OABC$ with respect to the reference configuration. Rotations $\theta _2$ and $\theta _4$ are always related to $\theta _3$ by the loop equations of quadrilateral $OABC$ .

(3) \begin{equation} \left \{{\begin{array}{*{20}c}{OA \cos \left ( \varphi _3 + \theta _3 \right ) + b\cos \left ( \varphi _2 + \theta _2 \right ) - CB \cos (\varphi _4 + \theta _4) - c = 0} \\[5pt]{OA \sin \left ( \varphi _3 + \theta _3 \right ) + b\sin \left ( \varphi _2 + \theta _2 \right ) - CB \sin (\varphi _4 + \theta _4) = 0} \\ \end{array}} \right. \end{equation}

where $b$ is the length of the payload and $c$ is the distance between cable pulleys.

If $\alpha _3$ and $\alpha _4$ are small, the variations in the lengths of sides $OA$ and $CB$ due to the lateral motion of the hooks are small. Length $OA$ is given by:

(4) \begin{equation} OA = L_{30}cos(\alpha _3)+L_{50}cos(\alpha _5) \end{equation}

The Taylor’s expansion shows that the lateral motion of the hook has a second-order effect on length $OA$ .

(5) \begin{align} OA = L_{30}\left ( 1-\frac{\alpha _3^2}{2} \right ) + L_{50} \left ( 1-\frac{\alpha _5^2}{2} \right ) \end{align}

(6) \begin{align} OA = (L_{30}+L_{50})\left ( 1- \frac{L_{30}}{2L_{50}}\alpha _3^2 \right ) \end{align}

A similar equation holds true for $CB$ :

(7) \begin{equation} CB = \left(L_{30} + L_{50}\right) \left( 1- \frac{L_{30}}{2L_{50}}\alpha _4^2 \right) \end{equation}

If the symmetry conditions are introduced in the loop equations, the following equations hold:

(8) \begin{equation} \theta _4=\theta _3 \end{equation}

(9) \begin{equation} \theta _2 = \frac{(L_{30} + L_{50})(\alpha _4^2 L_{30}^2 - \alpha _3^2 L_{30}^2)}{2bL_{30}L_{50}} \end{equation}

It is worth noticing that the payload rotation is a second-order term.

The equations of motion of the cable system are developed with the Lagrange’s approach.

The system kinetic energy ( $E_k$ ) is equal to:

(10) \begin{equation} E_k = \frac{1}{2}m_3\left ({\dot x_3 ^2 + \dot y_3 ^2 } \right ) + \frac{1}{2}m_4\left ({\dot x_4 ^2 + \dot y_4 ^2 } \right ) + \frac{1}{2}M\left ({\dot x_G ^2 + \dot y_G ^2 } \right ) \end{equation}

The velocities of masses are calculated using first-order approximations:

(11) \begin{equation} \left \{{\begin{array}{*{20}c}{\dot x_3 = L_{30} \dot \alpha _3 + L_{30} \dot \theta _3 + \dot x_b } \\[4pt]{\dot y_3 = 0} \\ \end{array}} \right. \end{equation}

(12) \begin{equation} \left \{{\begin{array}{*{20}c}{\dot x_4 = L_{30} \dot \alpha _4 + L_{30} \dot \theta _3 + \dot x_b } \\[4pt]{\dot y_4 = 0} \\ \end{array}} \right. \end{equation}

(13) \begin{equation} \left \{{\begin{array}{*{20}c}{\dot x_G = \left ({L_{30} + L_{50} } \right )\dot \theta _3 + \dot x_b } \\[4pt]{\dot y_G = 0} \\ \end{array}} \right. \end{equation}

The springs and the dampers that represent the stiffness and the damping properties of the human arm have the first end-point fixed to the body and the second end-point fixed to the center of the orthosis (point G). The elastic potential energy ( $E_{p,el}$ ) is given by:

(14) \begin{equation} E_{p,el} = \frac{1}{2}k_x \left ({x_G - \frac{b}{2}} \right )^2 + \frac{1}{2}k_y \left ( y_G + (L_{30} + L_{50}) \right )^2 \end{equation}

where $k_x$ and $k_y$ are the horizontal and vertical arm stiffness; the gravity potential energy ( $E_{p,g}$ ) is given by:

(15) \begin{equation} E_{p,g} = m_3gy_3 + m_4gy_4 + Mgy_G \end{equation}

where the coordinates of the masses are given by:

(16) \begin{equation} y_3 = L_{30} sin \left ({3\pi }/{2} + \theta _3 + \alpha _3 \right ) = - L_{30} cos(\theta _3 + \alpha _3) \end{equation}

(17) \begin{equation} y_4 = L_{30} sin \left ({3\pi }/{2} + \theta _3 + \alpha _4 \right ) = - L_{30} cos(\theta _3 + \alpha _4) \end{equation}

(18) \begin{equation} \begin{cases} x_G = OA cos \left ({3\pi }/{2} + \theta _3 \right ) + x_b + \dfrac{b}{2}cos \left ( \theta _2 \right )= (L_{30} + L_{50}) \left ( 1 - \dfrac{L_{30}}{2L_{50}} \alpha _{3}^{2} \right ) sin \theta _3 + x_b + \dfrac{b}{2} \\[15pt] y_G = OA sin \left ({3\pi }/{2} + \theta _3 \right ) + \dfrac{b}{2} sin\left ( \theta _2 \right ) = - (L_{30} + L_{50}) \left ( 1- \dfrac{L_{30}}{2L_{50}}\alpha _3^2 \right ) cos \theta _3 + \dfrac{\left ( L_{30} + L_{50} \right )\left (\alpha _4^2 - \alpha _3^2 \right ) L_{30}}{4L_{50}} \end{cases} \end{equation}

If second-order Taylor’s expansions of the trigonometric functions are adopted, the coordinates needed for the calculations of the potential energy are given by the following equations:

(19) \begin{equation} y_3 = -L_{30} \left ( 1 - \frac{(\theta _3 + \alpha _3)^2}{2} \right ) \end{equation}

(20) \begin{equation} y_4 = -L_{30} \left ( 1- \frac{(\theta _3 + \alpha _4)^2}{2} \right ) \end{equation}

(21) \begin{equation} \begin{cases} x_G = \left ( L_{30} + L_{50} \right )\theta _3 + x_b + \dfrac{b}{2}\\[10pt] y_G = - \left ( L_{30} + L_{50} \right ) \left ( 1 - \dfrac{\theta _3^2}{2} - \dfrac{L_{30}}{2L_{50}} \alpha _3^2 \right ) + \dfrac{\left (L_{30} + L_{50} \right ) \left ( \alpha _4^2 - \alpha _3^2 \right ) L_{30}}{4L_{50}} \end{cases} \end{equation}

where in Equation 21, the term proportional to $\alpha _3^2\theta _3^2$ has been neglected since it is a fourth-order term.

Using the Lagrange’s approach, the equations of forced damped vibrations in matrix form are as follows:

(22) \begin{equation} {\bf M}_{\bf s} \cdot \ddot{\mathbf{q}} +{\bf C}_{\bf s} \cdot \dot{\mathbf{q}} +{\bf K}_{\bf s} \cdot \mathbf{q} = \mathbf{T} \end{equation}

where ${\bf M}_{\bf s}$ is the mass matrix and ${\bf C}_{\bf s}$ is the damping matrix, whereas $\mathbf{T}$ is the forcing torque and $\mathbf{q}=\{\theta _3,\alpha _3,\alpha _4\}^T$ . Since the stiffness matrix ${\bf K}_{\bf s}$ includes both the elastic and the gravitational terms, two matrices $K_e$ and $K_g$ are introduced to represent the elastic and gravitational parts, respectively. The mass, damping and the two stiffness matrices and the forcing torque of the model have the following form:

(23) \begin{equation} \begin{array}{*{20}c}{{\bf M}_{\bf s} = \left [{\begin{array}{*{20}c}{L_{30} ^2 \left ({m_3 + m_4 } \right ) + M\left ({L_{30} + L_{50} } \right )^2 } & \quad {L_{30} ^2 m_3 } & \quad {L_{30} ^2 m_4 } \\[5pt]{L_{30} ^2 m_3 } & \quad {L_{30} ^2 m_3 } & \quad 0 \\[5pt]{L_{30} ^2 m_4 } & \quad 0 & \quad {L_{30} ^2 m_4 } \\[5pt] \end{array}} \right ]} & \quad {{\bf C}_{\bf s} = \left [{\begin{array}{*{20}c}{c_x\left ( L_{30} + L_{50} \right )^2} & \quad 0 & \quad 0 \\[5pt] 0 & \quad 0 & \quad 0 \\[5pt] 0 & \quad 0 & \quad 0 \\[5pt] \end{array}} \right ]} \\[5pt] \end{array} \end{equation}

(24) \begin{equation}{\bf K}_{\bf e} = \left [{\begin{array}{*{20}c}{k_x\left ({L_{30} + L_{50} } \right )^2} & \quad 0 & \quad 0 \\[5pt] 0 & \quad 0 & \quad 0 \\[5pt] 0 & \quad 0 & \quad 0 \\[5pt] \end{array}} \right ] \end{equation}

(25) \begin{equation}{\bf K}_{\bf g} = \left [{\begin{array}{*{20}c}{g\left [{L_{30} \left ({m_3 + m_4 } \right ) + M\left ({L_{30} + L_{50} } \right )} \right ] } & \quad {m_3 gL_{30} } & \quad {m_4 gL_{30} } \\[5pt]{m_3 gL_{30} } & \quad {\dfrac{{\left ({2L_{50} m_3 + M\left ({L_{30} + L_{50} } \right )} \right )L_{30} g}}{{2L_{50} }}} & \quad 0 \\[5pt]{m_4 gL_{30} } & \quad 0 & \quad {\dfrac{{\left ({2L_{50} m_4 + M\left ({L_{30} + L_{50} } \right )} \right )L_{30} g}}{{2L_{50} }}} \\[5pt] \end{array}} \right ] \end{equation}

(26) \begin{equation}{\bf T} = - \left \{{\begin{array}{*{20}c}{\left [{L_{30} \left ({m_3 + m_4 } \right ) + M\left ({L_{30} + L_{50} } \right )} \right ]\ddot x_b + c_x\left ({L_{30} + L_{50} } \right )\dot x_b + k_x\left ({L_{30} + L_{50} } \right )x_b } \\[5pt]{L_{30} m_3 \ddot x_b } \\[5pt]{L_{30} m_4 \ddot x_b } \\[5pt] \end{array}} \right \} \end{equation}

Substituting Equation (21) into Equation (14), the difference $(y_G + (L_{30} + L_{50}))$ leads to fourth-order terms in $k_y$ , which are neglected in $K_e$ . A proportional damping assumption is made [Reference Rao24] and the damping matrix is assumed proportional to $K_e$ . It is worth noticing that the matrices are constant and the equations of motion describing the forced vibrations due to robot arm motion are a system of linear coupled differential equations.

The natural frequencies and the modes of vibration of this model are calculated solving the eigenvalue problem. The first mode is a pendulum mode of the whole system, whereas the second and third modes are the hook modes dominated by the transverse vibrations of the hook; they are similar to the modes found in [Reference Zuccon, Doria, Bottin and Rosati36] where a 5 DOF model of the free vibrations of a cable robot was developed.

As reported in Table II, very different values of the human arm stiffness $k_h$ and damping factor $c_h$ are present in the literature. The influence of arm stiffness on the natural frequencies of the modes of vibration is depicted in Fig. 6, considering the parameters of the planar model reported in Table III. The natural frequency of the pendular mode increases from 0.72 Hz (when arm stiffness is 0) to 1.48 Hz when the arm stiffness reaches the maximum value of Table II. Conversely, the variations in the natural frequencies of the 2 $^{nd}$ and 3 $^{rd}$ modes are negligible.

Table III. Parameters of the mathematical model for the transverse forced vibration due to robot arm motion.

Figure 6. Influence of the human arm stiffness $k_x$ on the natural frequency of first transverse mode (pendulum mode) (a) and of the second and third modes (hook modes) (b).

3.2. Response to robot arm motion

The simulated input motion of the robot $x_b$ is a sinusoidal signal:

(27) \begin{equation} x_b(t) = x_0 sin(\omega t) \end{equation}

with amplitude $x_0$ set to 0.150 m and angular frequency $\omega$ set to 0.6283 rad/s, corresponding to a frequency of 0.1 Hz. The reason for choosing such input motion is twofold:

• It is a realistic input;

• Since the system is linear, harmonic analysis can be used to study the effect of more complex inputs (e.g., periodic inputs)

Figure 7 deals with the forced vibration due to the robot motion considering four different models of arm stiffness and damping. Figure 7a and b represents the amplitude of the forcing torque $T$ for each DOF. Figure 7c and d represents the time histories of $\theta _3$ and $\alpha _3$ , respectively. $\alpha _4$ is not represented since it behaves as $\alpha _3$ (since the only difference in the equations of motion disappears due to the assumption $m_3=m_4$ ).

Figure 7. Transverse forced vibration due to robot motion for four values of stiffness and damping of patient’s arm: (a) amplitude of forcing $T_{\theta _3}$ for $\theta _3$ ; (b) amplitude of forcing $T_{\alpha _3}$ for $\alpha _3$ ; (c) time history of $\theta _3$ ; (d) time history of $\alpha _3$ .

The first stiffness and damping values are the ones from Doria et al. [Reference Doria, Tognazzo and Cossalter11] and from Dyck et al. [Reference Dyck and Tavakoli12], since they present maximum stiffness (Doria et al.) and minimum stiffness (Dyck et al.), respectively. The third set of values includes the average values of $k_h$ and $c_h$ ( $k_{avg}$ = 78.66 N/m and $c_{avg}$ = 7.40 Ns/m). In the fourth model, the stiffness and damping values of the arm and forearm are set to zero, representing the absence of any reactions of the human arm.

The forcing torques are harmonic functions. The forcing torque $T_{\theta _3}$ on $\theta _3$ largely depends on arm stiffness and damping. In fact, for $k_h$ and $c_h$ = 0 the value is very small (in Fig. 7a it is amplified by a factor of 20). Moreover, since in this case $T_{\theta _3}$ depends only on $\ddot{x}_b$ , it is in opposition with respect to the torques referring to the other cases. The amplitude of the forcing torque $T_{\alpha _3}$ acting on $\alpha _3$ is very small and constant, since Equation (26) states that it depends only on $L_3$ , $m_3$ and on the acceleration of the robot $\ddot{x}_b$ .

The time domain response does not highlight resonance phenomena. $\theta _3$ chiefly oscillates at the forcing frequency (0.1 Hz). When arm damping is small, small amplitude transient vibrations at the frequency of the pendular mode appear as well. The response of $\alpha _3$ depicted in Fig. 7d is characterized by the presence of vibrations at the forcing frequency. There are small high-frequency vibrations at the natural frequency of the hook modes. When stiffness is large and damping is small, there are some vibrations at the frequency of the pendular mode as well. The amplitude of the main harmonic at the forcing frequency decreases if arm stiffness decreases and becomes very small when $k_h$ and $c_h$ are null (the corresponding curve in Fig. 7d is multiplied by a factor of 20). It is worth noticing that this phenomenon happens because, even if torque $T_{\alpha _3}$ does not depend on the stiffness and damping of the human arm, the equations of motion (22) are coupled by the mass and stiffness matrices.

These results are confirmed when the responses are analyzed in the frequency domain (Fig. 8). The main peak of the spectrum appears at the forcing frequency, and the peaks at the natural frequencies of the pendular mode and of the hook modes are very small.

Figure 8. Transverse forced vibration due to robot motion: (a) FFT of $\theta _3$ for the four cases considered and (b) of $\alpha _3$ for the four cases considered.

4.1. Model description

The two-cable model can be used to study the transverse forced vibrations due to cable motions having amplitude $\delta$ . In this case, the source of excitation is the periodic variation in the length of the upper cables, whereas the robot is considered steady ( $x_b = 0$ ), as shown in Fig. 9. In typical rehabilitation exercises, the cables move with the same motion law to keep the orthosis horizontal [Reference Rosati, Gallina and Masiero28]; therefore, in the model $L_4(t) = L_3(t)$ . Kinematic equations (1-9) and (19-21) dealing with the position of the masses and the rotations of the cables still hold true if $L_{30}$ is replaced by $L_3(t)$ and if $x_b$ is set to zero.

Figure 9. Model of transverse forced vibration due to cables’ elongation. The input is represented in red, and the DOFs $\theta _3$ , $\alpha _3$ , and $\alpha _4$ are represented in blue.

The equations of the elastic potential energy (14) and gravity potential energy (15) still hold true, but it is assumed that the human spring is unloaded when the orthosis is at the lowest height.

The equation of the kinetic energy (10) still holds true, but new expressions of velocity are needed to take into account the elongation velocity of the cables ( $\dot L_4(t) = \dot L_3(t)$ ). The hooks have both elongation velocity ( $\dot L$ ) and tangential velocity,

(28) \begin{equation} \begin{cases} \dot x_G = \dot L_3 \theta _3 + \left ({L_{3} + L_{50} } \right )\dot \theta _3 - \dfrac{1}{2} b \theta _2 \dot \theta _2 \\[9pt] \dot y_G = \theta _3 \left ( L_3 + L_{50} \right ) \dot \theta _3 - \dfrac{1}{2} \dot L_3 + \dfrac{1}{4} \left ( -2\dot L_3 + 2 \theta _3^2 \dot L_3 \right ) + \dfrac{2\alpha _4L_3(L_3+L_{50})\dot \alpha _4 + \alpha _4^2L_3\dot L_3 + \alpha _4^2\left ( L_3 + L_{50} \right ) \dot L_3}{4L_{50}} \\[17pt] \quad \quad \quad +\dfrac{2\alpha _3L_3(L_3+L_{50})\dot \alpha _3 + \alpha _3^2L_3\dot L_3 + \alpha _3^2\left ( L_3 + L_{50} \right ) \dot L_3}{4L_{50}} \\[5pt] \end{cases} \end{equation}

(29) \begin{equation} \left \{{\begin{array}{*{20}c}{\dot x_3 = L_{3} \left ( \dot \alpha _3 + \dot \theta _3 \right ) cos\left ( \alpha _3 + \theta _3 \right ) + \dot L_3 sin\left ( \alpha _3 + \theta _3 \right )} \\[5pt]{\dot y_3 = L_{3} \left ( \dot \alpha _3 + \dot \theta _3 \right ) sin\left ( \alpha _3 + \theta _3 \right ) - \dot L_3 cos\left ( \alpha _3 + \theta _3 \right )} \end{array}} \right. \end{equation}

(30) \begin{equation} \left \{{\begin{array}{*{20}c}{\dot x_4 = L_{3} \left ( \dot \alpha _4 + \dot \theta _3 \right ) cos\left ( \alpha _4 + \theta _3 \right ) - \dot L_3 sin\left ( \alpha _4 + \theta _3 \right )} \\[5pt]{\dot y_4 = L_{3} \left ( \dot \alpha _4 + \dot \theta _3 \right ) sin\left ( \alpha _4 + \theta _3 \right ) + \dot L_3 cos\left ( \alpha _4 + \theta _3 \right )} \end{array}} \right. \end{equation}

The equations of motions of forced damped vibrations in matrix form are derived using the Lagrange’s approach and neglecting third (and higher) order terms. They have the form of Equation (22). Again the stiffness matrix $K_S$ is split into two matrices $K_e$ and $K_g$ including the elastic and gravitational terms, respectively. The mass, damping, and stiffness matrices and the forcing torque of the model have the following forms:

(31) \begin{equation} \begin{array}{*{20}c}{{\bf M}_{\bf s} = \left [{\begin{array}{*{20}c}{L_{3} ^2 \left ({m_3 + m_4 } \right ) + M\left ({L_{3} + L_{50} } \right )^2 } & \quad {L_{3} ^2 m_3 } & \quad {L_{3} ^2 m_4 } \\[9pt]{L_{3} ^2 m_3 } & \quad {L_{3} ^2 m_3 } & \quad 0 \\[9pt]{L_{3} ^2 m_4 } & \quad 0 & \quad {L_{3} ^2 m_4 } \\[9pt] \end{array}} \right ]} \end{array} \end{equation}

(32) \begin{equation} \begin{array}{*{20}c}{{\bf C}_{\bf s} = \left [{\begin{array}{*{20}c}{c_x\left ( L_{30} + L_{50} \right )^2 - c_y\left (L_3 + L_{50} \right )\left ( L_3 - L_{30} - \delta \right )}& \quad\!\!\!\! 0 & \quad\!\!\!\! 0 \\[9pt] 0 & \quad\!\!\!\! \dfrac{-c_y\left (L_3+L_{50}\right ) \left ( L_3 - L_{50} - \delta \right )L_3}{{2L_{50} }} & \quad\!\!\!\! 0 \\[9pt] 0 & \quad\!\!\!\! 0 & \quad\!\!\!\! \dfrac{-c_y\left (L_3+L_{50}\right ) \left ( L_3 - L_{50} - \delta \right )L_3}{{2L_{50} }} \\[9pt] \end{array}} \right ]} \\[9pt] \end{array} \end{equation}

(33) \begin{equation}{\bf K}_{\bf g} = \left [{\begin{array}{*{20}c}{g\left [{L_{3} \left ({m_3 + m_4 } \right ) + M\left ({L_{3} + L_{50} } \right )} \right ]}& \quad\!\!\!\! {m_3 gL_{3} } & \quad\!\!\!\! {m_4 gL_{3} } \\[9pt]{m_3 gL_{3} } & \quad\!\!\!\! {\dfrac{{\left ({2L_{50} m_3 + M\left ({L_{3} + L_{50} } \right )} \right )L_{3} g}}{{2L_{50} }}} & \quad\!\!\!\! 0 \\[9pt]{m_4 gL_{3} } & \quad\!\!\!\! 0 & \quad\!\!\!\! {\dfrac{{\left ({2L_{50} m_4 + M\left ({L_{3} + L_{50} } \right )} \right )L_{3} g}}{{2L_{50} }}} \\[9pt] \end{array}} \right ] \end{equation}

(34) \begin{equation}{\bf K}_{\bf e} = \left [{\begin{array}{*{20}c}{k_x\left ({L_{3} + L_{50} } \right )^2 -k_y\left (L_3 + L_{50} \right )\left ( L_3 - L_{30} - \delta \right )} & \quad\!\!\!\! 0 & \quad\!\!\!\! 0 \\[9pt] 0 & \quad\!\!\!\! {\dfrac{-k_y\left (L_3+L_{50}\right ) \left ( L_3 - L_{30} - \delta \right )L_3}{{2L_{50} }}} & \quad\!\!\!\! 0 \\[9pt] 0 & \quad\!\!\!\! 0 & \quad\!\!\!\! {\dfrac{-k_y\left (L_3+L_{50}\right ) \left ( L_3 - L_{30} - \delta \right )L_3}{{2L_{50} }}} \\[9pt] \end{array}} \right ] \end{equation}

(35) \begin{equation}{\bf T} = - \left \{{\begin{array}{*{20}c}{2{L_{3} \dot L_3 \left ( m_3 \left ( \dot \alpha _3 + \dot \theta _3 \right )+ m_4 \left ( \dot \alpha _4 + \dot \theta _3 \right )\right ) + 2M\left ( L_{3} + L_{50} \right ) \dot L_3 \dot \theta _3 }} \\[12pt]{2m_3L_3\dot L_3\left ( \dot \alpha _3 + \dot \theta _3 \right ) - \dfrac{1}{2}\dfrac{M\left (L_3 + L_{50}\right )\alpha _3 L_3 \ddot L_3}{L_{50}}}\\[12pt]{2m_4L_3\dot L_3 \left ( \dot \alpha _4 + \dot \theta _3 \right ) - \dfrac{1}{2}\dfrac{M\left (L_3 + L_{50}\right )\alpha _4L_3\ddot L_3}{L_{50}}} \\[5pt] \end{array}} \right \} \end{equation}

Due to the variation in cable lengths, the matrices of the systems depend on the configuration. It is worth noticing that the forcing torque on $\theta _3$ includes non-linear Coriolis terms, whereas the forcing torques on $\alpha _3$ and $\alpha _4$ include both Coriolis terms and terms related to cable acceleration. All the forcing terms depend either on $\dot \theta _3$ , $\dot \alpha _3$ , $\dot \alpha _4$ or on $\alpha _3$ , $\alpha _4$ . Hence, a non-null initial condition is needed to initiate the forced vibrations.

The natural frequencies and the modes of vibrations of this model are calculated by solving the eigenvalue problem for assigned configurations (cable lengths). Figure 10 shows the effect of cable length ( $L_4 = L_3$ ) on the natural frequencies of the transverse modes of vibration. It is interesting to note that the natural frequencies of the second and third transverse mode $f_{n2}$ and $f_{n3}$ assume the same value regardless of $k_h$ for a certain cable length, which corresponds to the lowest position of the orthosis (where the cable length is maximum). Indeed, in that point $L_3 = L_{30} + \delta$ , meaning that the terms in $k_y$ in the elastic part $K_e$ of the stiffness matrix are null.

Figure 10. Influence of the human arm stiffness $k_h$ on the natural frequency of first (a), second (b), and third (c) transverse mode.

On the other hand, the natural frequencies of the first transverse mode $f_{n1}$ for different values of $k_h$ do not reach a common value, but change during time keeping different values. This is reasonable as the first term of the elastic part $K_e$ of the stiffness matrix presents a component ( $k_x\left (L_3 + L_{50}\right )^2$ ) which does not become zero during the cycle.

Figure 11 represents influence of $L_3$ on the natural frequencies of the transverse modes. All the natural frequencies decrease as cable length increases. The natural frequencies of the second and third transverse modes (hook modes) assume the same value for the maximum elongation regardless of the human arm stiffness $k_h$ . On the other hand, the curves representing the value of the natural frequency of the first transverse mode for different values of $k_h$ do not intersect. It is worth noticing that the values of the natural frequencies for null values of the human arm stiffness and for $L_3 = 0.380$ m are equal to the ones obtained in [Reference Zuccon, Doria, Bottin and Rosati36].

Figure 11. Influence of the cable length $L_3$ on the natural frequency of first transverse mode (a), of the second one (b), and of the third one (c) for different human arm stiffness $k_h$ .

4.2. Response to cable motion

The simulated input motion of the cables is sinusoidal:

(36) \begin{equation} L_3(t) = L_{30}+\delta \sin (\omega _Lt) \end{equation}

amplitude $\delta$ is set to 0.150 m and angular frequency $\omega _L$ to 0.6283 rad/s, corresponding to 0.1 Hz. Initial cable length $L_{30}$ is set to 0.345 m.

Figure 12 deals with the forced vibration due to cable motion considering different models of arm stiffness and damping. Differently from Section 3.2, only two stiffness and damping models are considered, that is, the model provided by Doria et al. [Reference Doria and Tognazzo10], which features large stiffness and minimum damping, and the model with null stiffness and damping. The results obtained considering the other models are deemed uninteresting since they show only an initial transient, due to the large damping values.

Figure 12. Transverse forced vibration due to cable motion considering two values of stiffness and damping of patient’s arm: (a) amplitude of forcing $T_{\theta _3}$ for $\theta _3$ ; (b) amplitude of forcing $T_{\alpha _3}$ for $\alpha _3$ ; (c) FFT of $F_{\theta _3}$ ; (d) FFT of $F_{\alpha _3}$ .

Moreover, only the forcing torques and the responses of $\theta _3$ and $\alpha _3$ are discussed, since $\alpha _4$ has almost the same behavior of $\alpha _3$ due to the symmetric configuration of the system. Figure 12a and b depict the forcing torques pf $\theta _3$ and $\alpha _3$ in the time domain, whereas Fig. 12c and d depict the same torques in the frequency domain. Differently from the previous scenario (Fig. 7), the forcing torques have complex waveforms with the presence of a beating phenomenon. This result can be explained by looking at the typical forcing term due to Coriolis acceleration, for example, the term

(37) \begin{equation} T_{\theta _3}^* = 2M(L_3(t) + L_{50})\dot L_3 \dot \theta _3 \end{equation}

in Equation (35).

If Equation (36) is inserted in (37), the following result holds:

(38) \begin{equation} T_{\theta _3}^* = 2M(L_3 + L_{50}) \delta \omega _L cos(\omega _L t) \dot \theta _3 + 2M \delta ^2 sin(\omega _L t)\omega _L cos(\omega _Lt)\dot \theta _3 \end{equation}

Since harmonic oscillations of $\theta _3$ at the natural frequency ( $\omega _n = 2\pi f_{n1}$ ) are expected

(39) \begin{equation} \theta _3 = \theta _{30}sin\omega _{n1}t \end{equation}

Equation (38) becomes:

(40) \begin{equation} T_{\theta _3}^* = 2M (L_{30}+L_{50})\delta \theta _{30}\omega _L\omega _{n1}cos(\omega _Lt)cos(\omega _{n1}t)+M\delta ^2\theta _{30}\omega _L\omega _{n1}sin(2\omega _Lt)cos(\omega _{n1}t) \end{equation}

where the second term has been modified using the double-angle formula. Then, the Werner’s formulae are used to transform the products of trigonometric functions into sums:

(41) \begin{equation} \begin{split} T_{\theta _3}^* &= 2M \left [ L_{30} + L_{50} \right ]\delta \theta _{30}\omega _L\omega _{n1} \frac{1}{2}\left [ cos((\omega _{n1}+\omega _L)t) + cos((\omega _{n1} - \omega _L)t) \right ] + \\ & + M\delta ^2\theta _{30}\omega _L\omega _{n1}\frac{1}{2}\left [ sin((\omega _{n1} + 2\omega _L)t) - sin((\omega _{n1} + \omega _L)t)\right ] \end{split} \end{equation}

Since in the present case $\omega _L \lt \lt \omega _{n1}$ , Equation (40) states that the terms $cos(\omega _Lt)$ and $sin(2\omega _Lt)$ modulate the input torque generating the beating phenomenon. Moreover, Equation (41) states that the modulated input torque consists of two harmonics, the former having frequency $\omega _{n1}-\omega _L$ , the latter having frequency $\omega _{n1}+\omega _L$ (if the modulation frequency is $2\omega _L$ the two frequencies are $\omega _{n1}-2\omega _L$ and $\omega _{n1}+2\omega _L$ ). This result is in agreement with the spectra depicted in Fig. 12c and d that show the presence of clusters of harmonics around the natural frequencies of the system. It is worth noticing that the behavior of the actual system is made more complex by some factors:

• the presence of damping;

• the coupling between the various DOFs;

• the variation of natural frequencies with cable length, which is a non-linear effect.

For the above-mentioned reasons, there are clusters of harmonics instead of two harmonics around the natural frequencies.

Figure 12 shows that the main effect of arm stiffness is the shift of the cluster of harmonics corresponding to $f_{n1}$ toward higher frequencies. Figure 12 highlights that the introduction of arm damping has a large effect on the torques on $\theta _3$ and $\alpha _3$ causing in the frequency domain a large reduction in the torque components having frequencies around $f_{n2}$ and $f_{n3}$ .

Figure 13. Transverse forced vibration due to cable motion for two values of stiffness and damping of patient’s arm: (a) time history of $\theta _3$ ; (b) time history of $\alpha _3$ ; (c) FFT of $\theta _3$ ; (d) FFT of $\alpha _3$ .

Figure 14. Comparison between the polynomial motion law and the sinusoidal one adopted in the previous sections.

Figure 15. Comparison of the responses due to robot arm motion for different input motions.

The responses of the system in the time and in the frequency domains are depicted in Fig. 13. The amplitude of the pendulum motion of the whole system (rotation $\theta _3$ ) is much smaller than the rotation of the upper cable caused by the lateral motion of the hook (rotation $\alpha _3$ ). If there is no arm damping, the vibrations are periodic and are characterized by beating phenomenon caused by the particular waveforms of the forcing torques. In the presence of arm damping, the oscillations quickly extinguish. In particular, the spectra depicted in Fig. 13c and d show that the high-frequency vibrations of the upper cables are more affected by arm damping than the low-frequency pendulum vibrations.