2.1. Full model

In the full model, the mass matrix of the robot is calculated by means of the Lagrangian approach. Both the kinetic energy due to the masses of the links and the kinetic energy due to the inertia tensors of the links [Reference Siciliano, Sciavicco, Villani and Oriolo19] are considered. The mass matrix can be calculated as:

(9) \begin{align} \boldsymbol{M}(\boldsymbol{q})=\sum _{i=1}^n \left ( m_{i} \boldsymbol{J}_P^{(i)T} \boldsymbol{J}_P^{(i)}+ \boldsymbol{J}_O^{(i)T} \boldsymbol{R}_i \boldsymbol{I}_{i} \boldsymbol{R}_i^T \boldsymbol{J}_O^{(i)} \right ) \end{align}

where $m_{i}$ and $\boldsymbol{I}_{i}$ represent the mass and the inertia tensor calculated at the center of mass (CoM) of the $i$ -th link, respectively, $\boldsymbol{J}_P^{(i)}$ and $\boldsymbol{J}_O^{(i)}$ are the two Jacobians that represent the relationships between the joint velocities and the linear and angular velocities of the $i$ -th link, respectively, $n$ is the number of links, and $\boldsymbol{R}_i$ is the rotation matrix from the reference frame of the $i$ -th link and the base frame.

The Jacobians are

(10) \begin{align} \boldsymbol{J}_P^{(l,i)} = \left [ \boldsymbol{j}_{{P}1}^{(l,i)} \quad \ldots \quad \boldsymbol{j}_{{Pi}}^{(l,i)} \quad \boldsymbol{0} \quad \ldots \quad \boldsymbol{0} \right ] \\[-28pt] \nonumber \end{align}

(11) \begin{align} \boldsymbol{J}_O^{(l,i)} = \left [ \boldsymbol{j}_{{O}1}^{(l,i)} \quad \ldots \quad \boldsymbol{j}_{{Oi}}^{(l,i)} \quad \boldsymbol{0} \quad \ldots \quad \boldsymbol{0} \right ] \\[0pt] \nonumber \end{align}

the columns of the matrices in (10) and (11) can be computed as follows:

(12) \begin{align} \boldsymbol{j}_{Pj}^{(l,i)} &= \begin{cases} \boldsymbol{z}_{j-1} & \quad \text{for a prismatic joint} \\ \boldsymbol{z}_{j-1} \times (\boldsymbol{p}_{\ell _i} - \boldsymbol{p}_{j-1}) & \quad \text{for a revolute joint} \end{cases} \\[-12pt] \nonumber \end{align}

(13) \begin{align} \boldsymbol{j}_{Oj}^{(l,i)} &= \begin{cases} \boldsymbol{0} & \quad \text{for a prismatic joint} \\ \boldsymbol{z}_{j-1} & \quad \text{for a revolute joint} \end{cases} \\[10pt] \nonumber \end{align}

where $\boldsymbol{p}_{j-1}$ and $\boldsymbol{z}_{j-1}$ are the position vector of the origin and the unit vector of the axis $z$ of the frame of the $(j-1)$ -th joint, respectively, and $\boldsymbol{p}_{\ell _i}$ is the position of the CoM of the $i$ -th link.

The full model is the most common model used in robotics to calculate the mass matrix of the system. However, the definition of parameters $m_{i}$ and $\boldsymbol{I}_{i}$ is usually difficult starting from the manuals and the datasheets given by the robot manufacturers. An estimation of such parameters can be obtained using CAD models [Reference Tonan, Bottin, Doria and Rosati20] or through extensive experimental tests [Reference Antonelli, Caccavale and Chiacchio21–Reference Dona’, Boscariol, Bottin, Lenzo and Rosati23].

2.2. Simplified model

The simplified model is an approximated model based on the approach presented in [Reference Cesaro, Bottin, Tonan, Doria and Rosati17]. It simplifies the mass distribution of the robot considering only two lumped masses for each link placed in such a way that the link mass $m_{i}$ and its CoM position are preserved, whereas the inertia tensor generally is not kept constant. The link reference frames are defined according to the Denavit–Hartenberg convention, and the lumped masses are placed along the $z_i$ and $z_{i+1}$ axes at appropriate distances from the link origins. It is worth noting that, although the link mass and the CoM position are derived from the CAD models, they are less influenced by the actual shape of the link internals, as the inertia tensor; hence, they are “robust” quantities.

In the simplified model, the number of lumped masses is equal to twice the number of links. Very often, some lumped masses are placed in the same position, thus they can be aggregated [Reference Cesaro, Bottin, Tonan, Doria and Rosati17].

The mass matrix of the simplified model can be calculated via the Lagrangian approach, in which only the linear velocities of the aggregated lumped masses have to be computed; this leads to a reduction in computational time [Reference Caneschi, Bottin, Doria, Cesaro and Rosati16, Reference Cesaro, Bottin, Tonan, Doria and Rosati17]. The mass matrix can be calculated as:

(14) \begin{align} \boldsymbol{M}(\boldsymbol{q})=\sum _{i=2}^N \left ( m_{l,i} \boldsymbol{J}_P^{(l,i)T} \boldsymbol{J}_P^{(l,i)} \right ) \end{align}

where $m_{l,i}$ is the $i$ -th aggregated mass, $\boldsymbol{J}_P^{(l,i)}$ is the Jacobian that links the joint velocities and the linear velocities of the $i$ -th aggregated mass, and $N \leq 2n$ is the total number of masses used to describe the system. Please note that the sum of Eq. (14) starts from 2 since one of the two masses of link 1 must be placed on the first axis, thus its linear velocity is zero.Footnote ²

Figure 1.

Position of the aggregated lumped masses (black spheres) in the robotic manipulator TechMan TM5-700.

Figure 1. Long description

A diagram of the position of aggregated lumped masses in the robotic manipulator TechMan TM5-700. The diagram is a 3D representation with axes labeled x, y, and z in millimeters. The robotic manipulator is depicted with several black spheres labeled as m_l1, m_l2, m_l3, m_l4, m_l5, m_l6, and m_l7, representing the positions of the aggregated lumped masses. The manipulator is shown in a vertical orientation with the base at the bottom and the end effector at the top. The diagram includes coordinate axes and dashed lines indicating the positions of the masses along the axes.

As an example, in Figure 1, the distribution of the aggregated lumped masses $m_{l,i}$ of a robotic manipulator, the TechMan TM5-700, is illustrated. The simplified model consists of seven masses, of which the first is steady since it is placed on the $z_1$ axis. Hence, the full model, which is made of 6 masses and 6 inertia tensors, is reduced to a simplified model made of only 7 lumped masses.

2.3. Standard model

The ISO 10218-2:2025 standard [6] establishes the safety requirements for the installation and use of industrial robots in collaborative applications. In fact, unlike traditional robots, which must be used within safety zones that separate the operator from the manipulator, collaborative robots are designed to allow safe interaction with people [Reference Di Cosmo, Giusti, Vidoni, Riedl and Matt24].

According to the ISO standard, the kinetic energy of the two impacting bodies before the impact can be equated to the elastic energy accumulated within the materials in contact during the impact:

(15) \begin{align} E_k=\frac {F^2}{2k}=\frac {1}{2} \mu v_{\mathit{rel}}^2 \end{align}

where $E_k$ represents the exchanged kinetic energy, $F$ is the elastic contact force, $k$ is the effective spring constant [7] of the materials in contact,Footnote ³ $v_{\mathit{rel}}$ is the relative velocity of the bodies along the normal to the plane tangent to the two colliding surfaces $\boldsymbol{n}$ , and $\mu$ is the reduced mass of the robot-operator system and is determined using the following formula:

(16) \begin{align} \mu = \left ( \frac {1}{m_H} + \frac {1}{m_R} \right )^{-1} \end{align}

Figure 2.

Simplified mass distribution model in the ISO/TS 15066:2016 standard [7].

in which $m_H$ represents the effective mass of the body part subjected to impact,Footnote ⁴ while $m_R$ represents the effective mass of the robot, calculated according to ISO 10218-2:2025 as follows:

(17) \begin{align} m_R = \frac {M}{2}+ m_L \end{align}

where $M$ is the total mass of the moving parts of the robot and $m_L$ is the payload, including the end-effector and the workpiece (Figure 2).

3.1. Lumped parameter model

Table I.

DH parameters of the links of the Omron i4-550L.

Figure 3.

Position of the aggregated lumped masses (black spheres) in the SCARA Omron i4-550L and a sample impact trajectory marked with a dashed black line.

The DH parameters are reported in Table I, while the controlled joint variables $q_1,\ldots ,q_4$ are represented in Figure 3 and collected in vector $\boldsymbol{q}$ . The mass and the position of the COM of each link have been estimated by means of a CAD software [Reference Tonan, Bottin, Doria and Rosati20]. In Table II, the total mass $m_i$ of each link and the values of the four parameters of the calculated lumped mass model are presented.

$O_i$ and $O_{i+1}$ are the origins of the frames of link $i$ and $i + 1$ , respectively. $L_{1i}$ is the position of the mass $m_{1i}$ measured from $O_i$ along the $z_i$ axis, while $L_{2i}$ is the position of the mass $m_{2i}$ measured from the intersection point between the $z_{i+1}$ and $x_i$ axes along the $z_{i+1}$ axis. The two quantities are defined according to the corresponding axis directions and can therefore assume both positive and negative values. The parameters $\kappa _{1i}=\frac {m_{1i}}{m_i}$ and $\kappa _{2i}=\frac {m_{2i}}{m_i}$ are the two normalized lumped masses defined such that their sum is 1.

The two masses located at the same point, but coming from the lumped parameter representation of two adjacent links, can be aggregated together. After this operation, the aggregated lumped masses are

(18) \begin{align} m_{l,1} &= \kappa _{11}m_1 \\[-12pt] \nonumber \end{align}

(19) \begin{align} m_{l,2} &= \kappa _{21}m_1 + \kappa _{12}m_2 \\[-12pt] \nonumber \end{align}

(20) \begin{align} m_{l,3} &= \kappa _{22}m_2 \\[-12pt] \nonumber \end{align}

(21) \begin{align} m_{l,4} &= \kappa _{13}m_3 + \kappa _{14}m_4 \\[-12pt] \nonumber \end{align}

(22) \begin{align} m_{l,5} &= \kappa _{23}m_3 + \kappa _{24}m_4 \\[10pt] \nonumber \end{align}

Table II.

Lumped parameters of the links of the Omron i4-550L.

Overall, the simplified model consists of 5 aggregated lumped masses as shown in Figure 3, the first is steady (as it belongs to the $z_1$ axis) and the other four are mobile. This is an important simplification because in this way the inertia model is composed of only 5 aggregated lumped masses instead of 4 masses and 4 inertia tensors.

3.2. Effective mass

We apply Equations from (3) to (8) to compute the effective mass for both the full and the simplified SCARA models. Before proceeding, the Jacobian matrix and the mass matrix are defined for each model.

The Jacobian matrix is evaluated at the impact point, located at a distance $R_x$ from the origin $O_4$ of the tool reference frame along the $x_4$ axis (Figure 4). The impact direction always coincides with $x_4$ .

Figure 4.

Impact point and its distance from the tool reference system.

The Jacobian matrix is the same for both models, and it is [Reference Hussain and Kanwal25]:

(23) \begin{align} \boldsymbol{J}(\boldsymbol{q}) = \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -l_1\sin (q_1) - l_2\sin (q_1 + q_2) + J_1 & -l_2\sin (q_1 + q_2) + J_1 & 0 & - J_1 \\[4pt] l_1\cos (q_1) + l_2\cos (q_1 + q_2) + J_2 & l_2\cos (q_1 + q_2) + J_2 & 0 & - J_2 \\[4pt] 0 & 0 & -1 & 0 \\[4pt] 0 & 0 & 0 & 0 \\[4pt] 0 & 0 & 0 & 0 \\[4pt] 1 & 1 & 0 & -1 \end{array}\right] \end{align}

where

$J_1 = - {R_x} \sin (q_1 + q_2 - q_4)$

$J_2 = {R_x} \cos (q_1 + q_2 - q_4)$

Planar impacts are considered. On the one hand, the orientation of the normal direction with respect to the fixed frame is $\phi$ . On the other hand, normal direction passing through the kinetic chain is $q_1 + q_2 - q_4$ . Therefore, the following orientation equation holds:

(24) \begin{align} \phi = q_1 + q_2 - q_4 \end{align}

and $q_4$ in Eq. (23) is not an independent variable.

The mass matrix depends on the inertia parameters and is therefore different for the two models. For the full model, the mass matrix is given by:

(25) \begin{align} \boldsymbol{M}(\boldsymbol{q}) = \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} M_{11}& M_{12} & 0 & -I_4 \\[4pt] M_{12} & M_{22} & 0 & -I_4 \\[4pt] 0 & 0 & m_3 + m_4 & 0 \\[4pt] -I_4 & -I_4 & 0 & I_4 \end{array}\right] \end{align}

where

$ M_{11} = m_1 r_1^2 + m_2(l_1^2 + r_2^2) + 2 m_2 l_1 r_2 \cos (q_2) + I_1 + I_2 + I_3 + I_4 + (m_3 + m_4)(l_1^2 + 2 l_1 l_2 \cos (q_2) + l_2^2)$

$M_{12} = m_2(r_2^2 + l_1 r_2 \cos (q_2)) + I_2 + I_3 + I_4 + (m_3 + m_4)(l_1 l_2 \cos (q_2) + l_2^2)$

$M_{22}=m_2 r_2^2 + I_2 + I_3 + I_4 + (m_3 + m_4) l_2^2$

and $l_i$ is the distance between origins $O_i$ and $O_{i+1}$ , and $r_i$ is the distance of the $i$ -th CoM to the origin $O_i$ .

In the simplified model, the mass matrix becomes a $3 \times 3$ matrix because, due to the placement of the lumped masses of the last link along the axis of the third joint (see Figure 3), the kinetic energy of the robot does not depend on $q_4$ . As a result, the Jacobian matrix should be considered with only three columns when computing the effective mass using Eq. (4). In this case, the mass matrix becomes

(26) \begin{align} \boldsymbol{M}(\boldsymbol{q}) = \left[\begin{array}{c@{\quad}c@{\quad}c} M_2 l_1^2 + M_3(l_1^2 + 2 l_1 l_2 \cos (q_2) + l_2^2) & M_3(l_1 l_2 \cos (q_2) + l_2^2) & 0 \\[4pt] M_3(l_1 l_2 \cos (q_2) + l_2^2) & M_3 l_2^2 & 0 \\[4pt] 0 & 0 & m_{l,4} + m_{l,5} \end{array} \right]\end{align}

where

$M_{2} = m_{l,2}$

$M_{3} = m_{l,3}+m_{l,4}+m_{l,5}$

Neglecting the motion along the $z$ axis, the behavior of the SCARA robot becomes that of a planar mechanism. Consequently, the aggregate masses $M_2$ and $M_3$ represent, respectively, the mass concentrated in the elbow and the mass concentrated in the end-effector of the equivalent planar mechanism (Figure 5). The mass $M_1 = m_{l,1}$ , positioned along the $z_1$ axis, is not considered in the calculation of the mass matrix $\boldsymbol{M}(\boldsymbol{q})$ , since it does not participate in the planar motion of the robot.

Given a normal direction $\boldsymbol{n}$ , the calculation of $m_n$ is performed using the Equations from (3) to (8). In the case here considered, the normal direction $\boldsymbol{n}$ lies in the horizontal plane, and the effective mass is always independent of $q_3$ , that is, of movement along the $z$ axis.

Figure 5.

Schematic SCARA model with impact point and normal direction $\boldsymbol{n}$ .

Figure 5. Long description

A schematic diagram of a SCARA model illustrating an impact point and normal direction. The diagram shows a robotic arm with three joints labeled M1, M2, and M3. The arm segments are labeled l1 and l2. The impact point is marked on the end effector, with the normal direction indicated by an arrow labeled n. The angles q1, q2, and q4 represent the joint angles, and the lengths l1 and l2 represent the arm segments. The coordinates x0, y0, x1, y1, x2, y2, x4, and y4 are marked at each joint and the impact point. The diagram also includes a coordinate system Rx and Ry at the impact point.

In the case here considered, the normal direction $\boldsymbol{n}$ , perpendicular to the plane tangent to the impact point, is aligned with the $x_4$ axis, since the contact surface is orthogonal to $x_4$ . If this condition does not hold, that is, if the surface is inclined with respect to the $x_4$ direction, then the inclination angle of the impact surface relative to $x_4$ must be taken into account in the orientation Eq. (24).

By substituting Eq. (24) into the Jacobian matrix, the effective mass $m_{n,f}$ for the full model becomes:

(27) \begin{align}& m_{n,f} \nonumber \\ & = \frac {-(M_{12}^2 - 2 I_4 M_{12} + I_4 M_{11} + I_4 M_{22} - M_{11} M_{22})}{ (M_{11} - 2 M_{12} + M_{22}) l_2^2 \sin ^2 \left ( q_1 + q_2 - \phi \right ) + (M_{22} - I_4) l_1^2 \sin ^2 \left ( q_1 - \phi \right ) + 2 (M_{22} - M_{12}) l_1 l_2 \sin \left ( q_1 + q_2 - \phi \right ) \sin \left ( q_1 - \phi \right ) } \end{align}

While for the simplified model the effective mass $m_{n,s}$ is

(28) \begin{align} m_{n,s} = \frac {M_3 \left ( M_3 \sin ^2(q_2) + M_2 \right )}{M_3 \sin ^2(q_2) + M_2 \sin ^2(q_1 + q_2 - \phi )} \end{align}

If the normal direction $\boldsymbol{n}$ coincides with the axes $\boldsymbol{x}$ or $\boldsymbol{y}$ , the calculus of the effective mass is even easier. For the $x$ -direction $\boldsymbol{n} = [1 \ 0 \ 0]^T$ , the rotation angle $\phi$ is equal to $0$ , while for the $y$ -direction $\boldsymbol{n} = [0 \ 1 \ 0]^T$ , the rotation angle $\phi$ is equal to $\frac {\pi }{2}$ .

For the ISO standard model, the effective mass $m_R$ is given, as mentioned in Eq. (17), by:

(29) \begin{align} m_R = \frac {M}{2} + m_L = \frac {\sum _i m_i}{2} + m_L \end{align}

where $m_L$ is the mass of the end-effector, since there is no workpiece.

4.1. Comparison of effective mass

Equation (28) shows that the effective mass depends only on $q_1$ and $q_2$ . The values of the aggregated masses $M_2$ and $M_3$ of the equivalent planar mechanism are 2.38 and 0.97 kg, respectively, while the joint ranges for the first two joints are $\pm$ 136 $^\circ$ and $\pm$ 148 $^\circ$ , respectively. The value of the mass for the ISO standard model is $m_R$ = 2.77 kg. Figure 6 shows the values of the effective mass given by the simplified model in the Cartesian and joint spaces when the impact direction coincides with the $x_0$ axis ( $\phi =0$ ). The red lines indicate the contours where the effective mass is equal to the ISO standard value $m_R$ .

Figure 6.

Values of the effective mass given by the simplified model in the Cartesian space (a) and in the joint space (b) with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ . The red curves represent the reduced mass calculated according to the ISO standard.

The plots were generated by linearly discretizing the ranges of the angles $q_1$ and $q_2$ into 50 intervals each. For improved visual interpretation, the data were represented using contour (isoline) regions. The two graphs show that the minima and maxima do not occur at isolated points, but extend along curved trajectories in the Cartesian and joint space. In particular, the minimum values of $m_{x,s}$ occur when $q_2 = -q_1 + \phi + \frac {\pi }{2}$ . By substituting this relation, it follows that $m_{n,s} = M_3$ . This particular condition corresponds to a specific configuration of the robot, in which the second link remains always perpendicular to the normal direction of impact. In this configuration, only the aggregated mass $M_3$ contributes to the impact as shown in Figure 7(a).

In contrast, the second link of the robot remains parallel to the direction of impact (see Figure 7(b)) when $q_2 = -q_1 + \phi$ . In this configuration, the effective mass reaches the maximum value, which is a function of $q_2$ :

(30) \begin{align} m_{n,s} = M_3 + \frac {M_2}{\sin (q_2)^2} \end{align}

Figure 7.

Schematic representation of the SCARA robot in configuration of minimum effective mass (a) and maximum effective mass (b) with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ and $\phi = 0$ .

If $\phi =0$ and $q_1=0$ , $q_2$ becomes $0$ and the robot reaches a singularity in the normal direction $\boldsymbol{n}$ , and, for this reason, the effective mass tends to infinity.

To further analyze the effect of robot configuration on the value of the effective mass, the diagram in Figure 8 shows the probability density of the values of the effective mass of the SCARA robot. The diagram is calculated considering 2500 robot configurations corresponding to 2500 end-effector locations uniformly distributed on the workspace. In the diagram, the mean, the median value, and the ISO standard value are reported as well. A bimodal distribution is observed. The value proposed by the ISO standard $m_R$ overestimates both the mean and the median. In particular, more than 70% of the values are less than the ISO standard value, highlighting the conservative nature of the ISO standard, which prioritizes safety.

Figure 8.

Probability density of the values of the effective mass in the simplified model with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ .

In order to compare the proposed simplified model with the full model, the relative errors in effective mass are calculated. The graphs in Figure 9 represent the percentage relative errors in effective mass ( $\Delta m_{n,s}$ ) in the Cartesian and joint space. It can be seen that the error in the worst case reaches 10%. However, in 75% of the cases, the error does not exceed 6%, indicating a good level of accuracy for the simplified lumped parameter model.

Figure 9.

Percentage relative error in effective mass between the full model and the simplified model in the Cartesian space (a) and joint space (b) with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ .

4.2. Comparison of reduced mass

The kinetic energy calculated by the ISO standard in Eq. (15) is the relative kinetic energy of the system before the impact. Introducing Eq. (16) in Eq. (15) the ISO standard energy can be expressed as a function of robot effective mass of the ISO standard $m_R$ and obstacle mass $m_H$ :

(31) \begin{align} E_{k,\text{ISO}} = \frac {1}{2}\frac {m_Rm_H}{m_R+m_H}(v_{1n}-v_{2n})^2 = \frac {1}{2}\mu _{\text{ISO}} v_{rel}^2 \end{align}

where $\mu _{\text{ISO}}$ is the reduced mass according to the ISO standard.

Figure 10.

Values of the reduced mass of the simplified model in the Cartesian space (a) and joint space (b) with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ for 4 different values of obstacle mass. The red curves represent the reduced masses calculated according to the ISO standard.

Figure 10. Long description

Panel A: A 3D contour plot shows the reduced mass in Cartesian space for four different obstacle masses. The x and y axes represent spatial coordinates in meters, while the vertical axis represents the reduced mass in kilograms. The plot includes four horizontal slices at different mass values, with color gradients indicating mass distribution. Red contours highlight specific mass levels. Panel B: Another 3D contour plot displays the reduced mass in joint space for the same obstacle masses. The x and y axes represent joint angles in degrees, and the vertical axis shows the reduced mass in kilograms. Similar to Panel A, this plot features four horizontal slices with color gradients and red contours marking mass levels. Both plots use a color scale ranging from dark blue to red to represent varying mass values.

In order to compare the ISO standard with both the full and the simplified models, the relative kinetic energy before impact can be calculated simply substituting in Eq. (31) $m_R$ with the effective masses predicted by the full and simplified models:

(32) \begin{align} E_{k,f} = \frac {1}{2}\frac {m_{n,f}m_H}{m_{n,f}+m_H}(v_{1n}-v_{2n})^2 = \frac {1}{2}\mu _{n,f} v_{rel}^2 \\[-28pt] \nonumber \end{align}

(33) \begin{align} E_{k,s} = \frac {1}{2}\frac {m_{n,s}m_H}{m_{n,s}+m_H}(v_{1n}-v_{2n})^2 = \frac {1}{2}\mu _{n,s} v_{rel}^2 \\[0pt] \nonumber \end{align}

where $m_{n,f}$ and $m_{n,s}$ are the effective mass and $\mu _{n,f}$ and $\mu _{n,s}$ the reduced masses for full and simplified models along the normal direction, respectively.

Since the kinetic energy depends only on the reduced mass and relative velocity, comparing the reduced masses is sufficient to evaluate the kinetic energies estimated by the two models. The relationship between the reduced mass and the obstacle mass is such that, when $m_{n,{f}}\to \infty$ (or $m_{n,{s}}\to \infty$ ), it follows that $\mu _{n,{f}} = m_H$ (or $\mu _{n,{s}} = m_H$ ). This is a relevant observation because, unlike the effective mass, the reduced mass never diverges.

A set of obstacle mass values was selected to analyze and represent the behavior of the reduced mass in both Cartesian and joint spaces. The selected obstacle masses are 0.5, 2.39, 3.72, and 7.35 kg, corresponding to the values used in the experimental tests. Figure 10 shows the values of the reduced mass calculated by means of the simplified model. The values of the ISO standard model $\mu _{\text{ISO}}$ (shown in red) corresponding to the four obstacle masses are 0.42, 1.28, 1.59, and 2.01 kg. The red lines in Figure 10 represent the contours where the reduced mass is equal to the ISO standard value $\mu _{\text{ISO}}$ . It can be observed that for small obstacle masses, the range of variation $\mu _{n,s}$ is narrower.

The graphs in Figure 11 represent the percentage relative errors in reduced mass ( $\Delta \mu _{n,s}$ ) between the full and simplified model in the Cartesian and joint space. It can also be seen that in the worst cases, the errors are at most 5%, highlighting the good approximation of the simplified lumped parameter model.

All the previously reported results dealt with impacts in $x$ direction. The analysis of the effect of impact direction was carried out considering the simplified model and $m_H = 2.39$ kg as a representative case. The values of the reduced masses $\mu _{n,s}$ for the normal direction $y$ and a generic direction in the Cartesian space are depicted in Figure 12.

Figure 11.

Percentage relative error in reduced mass between the full model and the simplified model in the Cartesian space (a) and joint space (b) with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ for 4 different values of obstacle mass.

Figure 12.

Values of the reduced mass of the simplified model in Cartesian space with $\boldsymbol{n} = [0 \ 1 \ 0]^T$ (a) and with $\boldsymbol{n} = [0.6 \ 0.8 \ 0]^T$ (b). The red curves represent the reduced mass calculated according to the ISO standard.

Figure 12. Long description

Panel A: A heat map showing the values of the reduced mass of the simplified model in Cartesian space. The x-axis is labeled x (m) and ranges from -0.5 to 0.5 meters. The y-axis is labeled y (m) and ranges from -0.5 to 0.5 meters. The color scale on the right represents the reduced mass in kilograms, ranging from 0.8 to 2.0. The red curves represent the reduced mass calculated according to the ISO standard. The heat map features concentric circles with varying colors, indicating different values of reduced mass. The central region is white, indicating the absence of data. Panel B: A heat map showing the values of the reduced mass of the simplified model in Cartesian space. The x-axis is labeled x (m) and ranges from -0.5 to 0.5 meters. The y-axis is labeled y (m) and ranges from -0.5 to 0.5 meters. The color scale on the right represents the reduced mass in kilograms, ranging from 0.8 to 2.0. The red curves represent the reduced mass calculated according to the ISO standard. The heat map features concentric circles with varying colors, indicating different values of reduced mass. The central region is white, indicating the absence of data.

The plots are equal to the one obtained for $\boldsymbol{n} = [1 \ 0 \ 0]^T$ , but are rotated in the Cartesian space by an angle corresponding to the rotation $\phi$ . As shown in Figure 13, rotating the impact direction by $90^\circ$ (from the $x$ direction to the $y$ direction) results in an equivalent configuration along the new normal direction, provided that the joint coordinate $q_1$ is also increased by $90^\circ$ . This implies that varying the orientation of the normal direction causes the effective mass values to rotate accordingly, producing plots that retain their shape but appear rotated in Cartesian space.

Figure 13.

Schematic representation of a SCARA robot with the same configuration according to the $x$ normal direction (a) and $y$ normal direction (b).

Finally, the effect of the left-arm configuration of the SCARA robot is considered. The results presented in Figure 14 show symmetry with respect to the normal direction, reflecting the intrinsic symmetry of the configuration.

Figure 14.

Values of the reduced mass of the simplified model in the Cartesian space (a) and joint space (b) with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ with left-arm configuration. The red curves represent the reduced mass calculated according to the ISO standard.

Figure 15.

Values of the minimum reduced mass of the simplified model in the Cartesian space with $\boldsymbol{n} = [1 \ 0 \ 0]^T$ in the two configurations. The red curves represent the reduced mass calculated according to the ISO standard.

Figure 15. Long description

A heat map represents the distribution of minimum reduced mass values in Cartesian space for two configurations. The heat map is circular with a central white area. The x-axis and y-axis are labeled in meters, ranging from -0.5 to 0.5. The color scale on the right ranges from 0.8 to 2.0 kilograms, with darker colors indicating lower values and lighter colors indicating higher values. Red curves represent the reduced mass calculated according to the ISO standard. The heat map shows a symmetrical pattern with higher values concentrated near the edges and lower values near the center.

Since there is an overlap between the workspaces attainable with left- and right-arm configurations, the configuration that minimizes the reduced mass can be chosen. Figure 15 represents the reduced mass in the Cartesian space, considering the minimum values calculated from the two configurations.