Subspace Projection-Based Collision Detection for Physical Interaction Tasks of Collaborative Robots

Abstract

In recent years, various collision detection methods have been proposed for safe human–robot collaboration; these methods are usually model-based. However, most tasks conducted by collaborative robots involve a physical robot–environment interaction, and it is very difficult to design a model of such interaction that varies depending on the external environment. Therefore, this study proposes a method for detecting collision between a human and a robot while the robot executes a task involving physical interaction with an unknown environment. To this end, a collision detection index is suggested based on the subspace projection technique. The proposed collision detection scheme is verified experimentally using a 7-DOF robot arm, and the experimental results show that collisions can be detected reliably by the proposed method.

1 Introduction

In many industries, collaboration between humans and robots is becoming increasingly important. However, collaborative robots could harm the humans with whom they share workspace, owing to human error or malfunction of the robot. Therefore, there has been a recent increase in the importance of human–robot collision safety. Collision detection is one of the most practical solutions to ensure human safety [1]. Various collision detection algorithms have been proposed to ensure the safety of human workers, one of which is a disturbance observer-based approach.

A disturbance observer, which is widely used to implement a robust controller, has been used to detect unwanted disturbances such as human–robot collision [2, 3] or actuator fault [4, 5]. These schemes observe an external torque estimated with the motor current and dynamic model of a robot and detect a collision if the external torque exceeds a certain threshold. Further researches were conducted to avoid the use of joint acceleration while estimating an external torque [6,7,8]. Concerning the model uncertainty, robust observers [9, 10] or neural network-based approaches [11, 12] have been proposed. Detection methods based on tactile sensors [13] or vision sensors [14] also have been proposed in order not to use manipulator’s dynamic model. However, there are several challenging issues for collision detection of a collaborative robot, one of which is collision detection during a physical robot–environment interaction.

Physical robot–environment interaction, such as handling of a payload and force control, is a fundamental function of collaborative robots, and the robots experience external force resulting from such interaction. However, aforementioned collision detection methods assume situations wherein the only source of the external force is the collision force. Thus, these schemes fail to detect collisions occurring along with an intentionally applied external force to perform the task, because these two external forces cannot be distinguished from one another. To address this problem, a collision detection algorithm that can reject the disturbance generated by the interaction force using a band-pass filter was proposed [15,16,17]. It assumes that the disturbance due to collisions occurs at a relatively higher frequency than that due to intentional contact to perform tasks such as kinesthetic teaching. However, the threshold frequency used to distinguish the collision from other disturbances is not clear, and there is no guarantee that the force resulting from the collision is always at a high frequency. To overcome these limitations, a collision detection method using tactile sensors with machine-learning algorithm was proposed [18]. However, this approach requires a vast amount of training data and time. Therefore, a more practical detection scheme should be developed for safe human–robot collaboration.

In this study, we propose a collision detection method for a collaborative robot that physically interacts with the external environment (objects in its surrounding). To achieve this, we devised a collision detection index (CDI) that is not affected by external forces exerted on the robot while performing normal tasks, and thus is sensitive only to collisions. This index is based on linear space projection. We conducted several experiments to validate the effectiveness of the proposed collision detection method.

The main features of the proposed method can be summarized in three points. First, it is possible to reliably detect collisions, even though the environment associated with the tasks is uncertain or unknown because the proposed method does not require any information about the environment. Second, collisions can be detected regardless of the frequency and magnitude of the collision force since the proposed CDI is fundamentally irrelevant to the external force generated during normal operation. Finally, it does not require any training data and pre-processing, such as learning and adaptation schemes.

The remainder of this paper is organized as follows. Section 2 introduces a collision detection scheme for a collaborative robot. Section 3 proposes the CDI, which is verified with several experiments detailed in Sect. 5. Finally, Sect. 6 presents the concluding remarks.

2 Collision Detection for Collaborative Robots

2.1 Classification of Collaborative Tasks

Generally, tasks that are conducted by a robot arm can be classified into three cases. The first is a non-interactive task such as robotic painting. In this case, the robot does not physically interact with objects in its surroundings, or the force exerted on the robot owing to the interaction between the robot and its surroundings is negligible. Figure 1a shows the external forces that can be applied to the robot when it performs a non-interactive task, and the only source of external force is the collision force F_c.

Fig. 1

Classification of human–robot collaboration tasks: a non-interactive task, b payload handing, and c contact task

The second case is based on the handling of a payload; a pick-and-place operation is a typical example. As shown in Fig. 1b, the external force F_p resulting from the grasped object or tool acts on the robot when it handles the payload. F_p is the resultant force of the gravitational force F_g and the inertial force F_i. Thus, both F_c and F_p are applied to the robot when a collision occurs. It is important to note that F_p, F_g, and F_i are vectors, and F_p is composed of a force vector f_p and a moment vector n_p. Likewise, F_i and F_g are denoted as (f_i, n_i) and (f_g, n_g), respectively.

The final case is a contact task, typical examples of which are deburring and assembly. In this case, F_c and the end-effector force F_e caused by the contact between the robot and the environment act on the robot, as shown in Fig. 1c. Therefore, F_c and F_e are exerted on the robot during collision.

2.2 External Torque-Based Collision Detection

The equation of motion can be described by using the external joint torque vector τ_ext, as follows

$${\varvec{\uptau}}_{j} = {\mathbf{M}}({\mathbf{q}}){\mathbf{\ddot{q}}} + {\mathbf{C}}({\mathbf{q}},{\dot{\mathbf{q}}})\,{\dot{\mathbf{q}}} + {\mathbf{g}}({\mathbf{q}}) + {\varvec{\uptau}}_{ext}$$

(1)

where M(q) is the inertia matrix, C(q,q̇) the matrix containing the Coriolis and centrifugal terms, and g(q) the gravity vector. The variables q, q̇, and q̈ are the joint position, velocity, and acceleration vectors, respectively. τ_j is the joint torque vector applied to each link of the robot arm and can be measured by joint torque sensors mounted at each joint for torque control.

The features of collaborative tasks are summarized in Table 1 in terms of τ_ext that has been used to detect collisions in the previous studies [2,3,4,5,6,7,8,9]. The only source of the external force is the collision force F_c when the robot conducts a non-interactive task. In such case, it is possible to establish τ_ext =  τ_c where τ_c is the external joint torque vector generated by F_c. This means that τ_ext = 0 if no collision occurs, and τ_ext ≠ 0 otherwise. On the contrary, τ_ext is affected by F_p or F_e when a robot handles a payload or conducts a contact task, and can be expressed as

$${\varvec{\uptau}}_{ext} = {\varvec{\uptau}}_{c} + {\varvec{\uptau}}_{p}$$

(2a)

$${\varvec{\uptau}}_{ext} = {\varvec{\uptau}}_{c} + {\varvec{\uptau}}_{e}$$

(2b)

where τ_p and τ_e are the joint torque vectors caused by F_p and F_e, respectively. As shown in (2), τ_c cannot be directly obtained from τ_ext because τ_ext ≠  τ_c. Therefore, a collision cannot be detected by observing τ_ext if a robot arm interacts with objects in its surroundings (i.e., τ_p ≠ 0 and/or τ_e ≠ 0).

Table 1 Main features of collaborative tasks

The collision detection indexes (CDIs) for each task are introduced in Table 1. The CDIs d_p and d_e are not affected by external forces exerted on the robot during payload handling or contact tasks, respectively. Therefore, they are sensitive only to collisions and can be used to detect a collision even when the contact force between the manipulator and the environment exists. These CDIs are derived and explained below in detail in Sect. 3.

If τ_p and τ_e are known, τ_c can be obtained from τ_ext; one method of obtaining τ_p and τ_e is to measure F_p and F_e using an F/T sensor mounted on the wrist of the arm. However, it is not economically feasible to use an expensive F/T sensor for collision detection only. An alternative method is to predict F_p and F_e based on the specific dynamic model of the grasped object and the static model of the environment in contact, respectively. However, it is very difficult to accurately develop or estimate such models since they vary according to the shape and material of the grasped object and the contact surface. Therefore, a collision detection method that does not require the use of the F/T sensor and the model of the external environment should be developed for safe physical robot–environment interaction.

3 Collision Detection Index

3.1 Collision Detection Index for Handling of Payload

To achieve the purpose of this study, subspace projections are used in the joint torque space. This is based on a well-known statement that a vector projected onto a vector space orthogonal to its own space always results in a zero vector. That is, if we project τ_ext in (2) onto the vector space orthogonal to the space wherein τ_p lies, then τ_p can be eliminated from τ_ext. By using this principle, the CDI d_p can be decoupled from F_p.

Projection of τ_ext onto the vector space orthogonal to the space wherein τ_p lies requires the projection matrix which is formulated from the linear mapping of vector spaces. Figure 2 shows the linear mapping between subspaces of joint torque space S_τ for an n-DOF manipulator. S_τ is an n-dimensional joint torque space, and τ_ext lies in this space. S_p is the space wherein τ_p lies, and it is a subspace of S_τ. ⊥S_p is the space orthogonal to S_p. The dimension of S_p and ⊥S_p are m and n-m, respectively. P_Sp and P_⊥Sp are the n × n projection matrices that project an arbitrary vector in S_τ onto spaces S_p and ⊥S_p, respectively. The dashed arrows in Fig. 2 represent the linear mapping.

Fig. 2

Mapping between the subspaces of joint torque space

As is well known, the projection matrix P_⊥Sp that projects τ_ext onto ⊥S_p is given by

$${\mathbf{P}}_{ \bot Sp} = \,{\mathbf{I}} - {\mathbf{P}}_{Sp}$$

(3)

where I is the n × n identity matrix. The projection matrix P_sp that projects an arbitrary vector onto S_p can be formulated by the well-known equation

$${\mathbf{P}}_{Sp} = {\mathbf{B}}_{p} \left( {{\mathbf{B}}_{p}^{T} {\mathbf{B}}_{p} } \right)^{ - 1} {\mathbf{B}}_{p}^{T}$$

(4)

where B_p is the basis matrix of S_p, and the columns of B_p are the orthogonal basis of S_p [19, 20]. Thus, B_p should be computed to derive P_⊥Sp.

To compute B_p, we use a basis of the space S_Fp, to which F_p belongs. τ_p and F_p are related by τ_p = J^TF_p as shown in Fig. 3a. Note that the basis of S_Fp is the set of a basis of the space S_fp and S_np wherein f_p and n_p lie, respectively, since F_p consists of f_p and n_p, which are linearly independent of each other, as shown in Fig. 1b. Thus, we find a basis of the space S_fp and S_np to compute B_p.

Fig. 3

Derivation of the projection matrix P_⊥Sp: a mapping between the joint torque space and task force space, and b plane of force generated by the payload

To find a basis of the space S_fp and S_np, F_p and its elements f_p and n_p were analyzed in terms of linear algebra. As shown in Fig. 3b, f_p lies in the same plane as f_g and f_i because F_p is a resultant of F_g and F_i, as shown in Fig. 1b where the unit vectors u_i and u_g are in the same directions as f_i and f_g, respectively. Then, we can consider the set of unit vectors u_i and u_g as a basis of S_fp that is the space wherein f_p lies because f_p is a linear combination of u_i and u_g. Therefore, matrix B_fp, which is an orthogonal basis of S_fp, can be established as follows:

$${\mathbf{B}}_{fp} = [\begin{array}{*{20}l} {{\mathbf{u}}_{i} } \hfill & {{\mathbf{u}}_{g} } \hfill \\ \end{array} ]$$

(5)

The magnitude of u_i and u_g are always known, because these vectors are unit vectors and the directions of u_i and u_g can be acquired from the desired acceleration of the end-effector and the gravitational force by assuming that the robot follows the desired trajectory well. That is, we can find u_i and u_g, which form B_fp, even though the exact magnitude and direction of f_p are unknown. Therefore, B_fp can be computed without the dynamic model of the grasped object.

On the contrary, matrix B_np, which is a basis of a space S_np wherein the vector n_p lies, cannot be found without the dynamic model of the grasped object, because the magnitude and direction of n_p are determined by the position of the center of mass of the grasped object, which is uncertain. This means that n_p can be any vector in the three-dimensional space. Therefore, B_np is a set of three linearly independent vectors and can be established as follows:

$${\mathbf{B}}_{np} = \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]$$

(6)

Therefore, the 6 × 5 matrix B_Fp, which is an orthogonal basis of S_Fp, is a set of B_fp and B_np. As a result, B_Fp can be established as follows:

$${\mathbf{B}}_{Fp} = \left[ {\begin{array}{*{20}l} {{\mathbf{B}}_{fp} } \hfill & {{\mathbf{0}}_{3 \times 3} } \hfill \\ {{\mathbf{0}}_{3 \times 2} } \hfill & {{\mathbf{B}}_{np} } \hfill \\ \end{array} } \right]$$

(7)

Then, the 6 × 6 projection matrix P_Fp, which projects an arbitrary force vector F into S_Fp, is given by substituting B_Fp into (4) as follows:

$${\mathbf{P}}_{Fp} = {\mathbf{B}}_{Fp} \left( {{\mathbf{B}}_{Fp}^{T} {\mathbf{B}}_{Fp} } \right)^{ - 1} {\mathbf{B}}_{Fp}^{T}$$

(8)

Therefore, as illustrated in Fig. 3a, τ_p can be represented by

$${\varvec{\uptau}}_{p} = {\mathbf{J}}^{T} {\mathbf{F}}_{p} = {\mathbf{J}}^{T} {\mathbf{P}}_{Fp} {\mathbf{F}} = {\mathbf{P}}'_{Sp} {\mathbf{F}}$$

(9)

where J is the 6 × n Jacobian matrix that can be derived from the manipulator kinematics and the matrix P’_Sp = J^TP_Fp, which is the n × 6 matrix. As can be seen from (9), τ_p is a linear combination of the column vectors of P’_Sp. That is, the column vectors of P’_Sp are a spanning set for S_p, where τ_p lies, and an orthogonal basis of S_p can be obtained from P’_Sp because the basis of a space is the particular spanning set for that space.

The singular value decomposition (SVD) method is known as the best way to construct the orthogonal basis of a space because the numerical computation of an SVD is relatively stable in terms of round-off errors compared to other orthogonalization methods. Therefore, SVD is employed to construct the orthogonal basis of S_p, and P’_Sp is decomposed as follows:

$${\mathbf{P}}'_{Sp} = {\mathbf{U\varSigma }}V^{T}$$

(10)

where U and V are the unitary matrices and Σ is the diagonal matrix. The column vectors of U and V are the bases for the row and column spaces of J_p, and the diagonal entries of Σ are the singular values of J_p. Therefore, B_p, the orthogonal basis of S_p, can be constructed using the column vectors of V, with the exception of the zero vectors of V.

By substituting B_p into (4), the projection matrix P_Sp can be derived as

$${\mathbf{P}}_{Sp} = {\mathbf{B}}_{p} ({\mathbf{B}}_{p}^{T} {\mathbf{B}}_{p} )^{ - 1} {\mathbf{B}}_{p}^{T} = {\mathbf{B}}_{p} {\mathbf{B}}_{p}^{ + }$$

(11)

where the superscript + denotes the pseudo-inverse operation defined by A⁺ = (A^TA)⁻¹A^T. By substituting (11) into (3), the projection matrix ${\mathbf{P}}_{ \bot Sp}$, which projects the arbitrary torque vector into ⊥S_p, can be derived as

$${\mathbf{P}}_{ \bot Sp} = {\mathbf{I}} - {\mathbf{B}}_{p} {\mathbf{B}}_{p}^{ + }$$

(12)

As mentioned, the two vector spaces S_p and ⊥S_p are orthogonal to each other, and τ_p always lies in S_p. Therefore, the projection of τ_p onto ⊥S_p is given by

$${\mathbf{0}} = ({\mathbf{I}} - {\mathbf{B}}_{p} {\mathbf{B}}_{p}^{ + } ){\varvec{\uptau}}_{p} \,$$

(13)

This means that τ_p cannot affect the projection of an arbitrary vector into ⊥S_p. As a result, a collision detection index d_p that is decoupled from τ_p can be developed by projecting τ_ext into ⊥S_p as follows:

$$\begin{aligned} {\mathbf{d}}_{p} & = ({\mathbf{I}} - {\mathbf{B}}_{p} {\mathbf{B}}_{p}^{ + } ){\varvec{\uptau}}_{ext} = ({\mathbf{I}} - {\mathbf{B}}_{p} {\mathbf{B}}_{p}^{ + } ){\varvec{\uptau}}_{c} + ({\mathbf{I}} - {\mathbf{B}}_{p} {\mathbf{B}}_{p}^{ + } ){\varvec{\uptau}}_{p} \\ & = ({\mathbf{I}} - {\mathbf{B}}_{p} {\mathbf{B}}_{p}^{ + } ){\varvec{\uptau}}_{c} \\ \end{aligned}$$

(14)

As shown in (14), d_p is decoupled from τ_p and is a function of τ_c and B_p. Therefore, if τ_c is applied to the robot arm, d_p increases or decreases with τ_c; d_p is close to zero if there are no collisions (i.e., τ_c = 0). As a result, collisions can be detected by observing d_p, even though the external force F_p is uncertain or unknown.

3.2 Collision Detection Index for Contact Task

Similar to the CDI for payload handling in the previous section, the CDI for the contact task is the result of projecting τ_ext onto the space orthogonal to τ_e. The projection matrix employed to project τ_ext onto the space orthogonal to τ_e is derived from the relation between the joint torque and the end-effector force as follows:

$${\varvec{\uptau}}_{e} = {\mathbf{J}}^{T} {\mathbf{F}}_{e}$$

(15)

In (15), τ_e is a linear combination of the column vectors of J^T, which are orthogonal to each other when the robot is not operated in a singular configuration. This means that a basis of the space S_e, where τ_e lies, is the set of the column vectors of J^T. Therefore, a projection matrix P_Se that projects an arbitrary vector onto S_e can be formulated by

$${\mathbf{P}}_{Se} = {\mathbf{J}}^{T} ({\mathbf{JJ}}^{T} )^{ - 1} {\mathbf{J}} = {\mathbf{J}}^{T} ({\mathbf{J}}^{T} )^{ + }$$

(16)

Then, the projection matrix P_⊥Se, which projects an arbitrary vector onto the space ⊥S_e, which is orthogonal to S_e, can be given by

$${\mathbf{P}}_{ \bot Se} = \,{\mathbf{I}} - {\mathbf{P}}_{Se}$$

(17)

Substituting (16) into (17) yields

$${\mathbf{P}}_{ \bot Se} = \,{\mathbf{I}} - {\mathbf{J}}^{T} ({\mathbf{J}}^{T} )^{ + }$$

(18)

As a result, we can project an arbitrary vector onto ⊥S_e using the projection matrix P_⊥Se in (18). The projection of τ_e onto ⊥S_e is given by

$${\mathbf{0}} = (\,{\mathbf{I}} - {\mathbf{J}}^{T} ({\mathbf{J}}^{T} )^{ + } ){\varvec{\uptau}}_{e}$$

(19)

This means that τ_e cannot affect the projection of an arbitrary vector onto ⊥S_e. Therefore, a collision detection index that is decoupled from τ_e can be developed by projecting τ_ext onto ⊥S_e as follows:

$$\begin{aligned} {\mathbf{d}}_{e} & = ({\mathbf{I}} - {\mathbf{J}}^{T} ({\mathbf{J}}^{T} )^{ + } ){\varvec{\uptau}}_{ext} \\ & = ({\mathbf{I}} - {\mathbf{J}}^{T} ({\mathbf{J}}^{T} )^{ + } ){\varvec{\uptau}}_{c} + ({\mathbf{I}} - {\mathbf{J}}^{T} ({\mathbf{J}}^{T} )^{ + } ){\varvec{\uptau}}_{e} \\ & = ({\mathbf{I}} - {\mathbf{J}}^{T} ({\mathbf{J}}^{T} )^{ + } ){\varvec{\uptau}}_{c} \\ \end{aligned}$$

(20)

As shown in (20), d_e is decoupled from τ_e, and becomes a function of τ_c and J. Therefore, collisions can be detected by observing d_e, even though the robot performs its task under the external force F_e.

4 Collision Detection Algorithm

Figure 4 shows the flowchart of the collision detection algorithm that was designed based on the proposed CDIs. This algorithm requires an estimate of τ_ext and then computes d_p and d_e. Ideally, every element of τ_ext, d_p and d_e approaches zero when no collision exists, but this condition rarely happens due to the uncertainty in the dynamic model and friction model. Therefore, collisions should be detected by comparing the absolute values of τ_ext, d_p and d_e with a thresholds τ_th, d_th,p and d_th,e, respectively. The suitable values of these thresholds are experimentally determined so that the CDIs exceeding these thresholds are considered as a collision. In Fig. 4, cases 1, 2, and 3 correspond to a non-interactive task, payload handling, and contact task in Fig. 1, respectively. Collision detection changes the CDI in accordance with the type of task conducted by the robot. If a collision is detected, the robot stops its task and regenerates its trajectory.

Fig. 4

Flowchart of the collision detection algorithm

Detection of a collision using the algorithm shown in Fig. 4 requires the value of τ_ext, which uses the acceleration information as shown in (1). However, the acceleration calculated from the encoder is susceptible to noise due to numerical differentiation. To solve this problem, we use a single typical observer for the estimation of τ_ext, which is designed based on the disturbance observer, and the generalized momentum p, which can be expressed as p= M(q)q̇. Using this observer, τ_ext can be estimated as follows:

$${\hat{\mathbf{\tau }}}_{ext} (t)\, = {\mathbf{K}}\left[ {\int_{0}^{t} {\left\{ {{\varvec{\uptau}}_{j} + {\mathbf{C}}^{T} ({\mathbf{q}},{\dot{\mathbf{q}}}){\dot{\mathbf{q}}} - {\mathbf{g}}({\mathbf{q}}) - {\hat{\mathbf{\tau }}}_{ext} (l)} \right\}} dl - {\mathbf{p}}} \right]$$

(21)

where $\hat{\varvec{\tau }}_{ext}$ is the observed value of τ_ext and K is the observer gain [6].

5 Experimental Verification

In this section, the CDIs proposed in the previous section are verified through several experiments. This verification was conducted using a 7-DOF robot arm (KU-Dex) that was developed in our laboratory. Every joint of this robot arm is equipped with a joint torque sensor (JTS), and its specifications are listed in Table 2.

Table 2 Specifications of KU-Dex for experimental verification

The proposed CDI for payload handling was experimentally verified. An arbitrary trajectory of the end-effector through P₁, P₂, and P₃ was given as shown in Fig. 5. This verification was conducted in three steps. Step 1 was conducted without the payload, and steps 2 and 3 were carried out with the payload.

Fig. 5

Trajectory of the end-effector for experimental verification

Step 1 of the experiment was conducted based on a situation in which the robot controlled its position along a given trajectory, as shown in Fig. 5. In this step, there was no physical interaction between the robot arm and the external environment; thus, F_p = 0. However, in step 2, the 1 kg mass shown in Fig. 6a was attached to the end- effector of the robot arm, as shown in Fig. 6b. In this step, the robot controlled its position in the same way as demonstrated in Fig. 5, and collisions between a human hand and the end-effector of the robot arm occurred while the robot was completing its task, as shown in Fig. 6b. Likewise, the 2 kg mass shown in Fig. 6a was attached to the end-effector in step 3, and the body of the robot collided with a human arm, as shown in Fig. 6c. Thus, both F_p and F_c were applied to the robot arm in steps 2 and 3 of this experiment.

Fig. 6

Experimental conditions for the verification of the CDI for handling of the payload: a two payloads, b collision with 1 kg payload, and c collision with 2 kg payload

Figure 7a, b show the values of $\hat{\varvec{\tau }}_{ext}$ at joint 4, which is mainly affected by F_p, and the fourth element of d_p during the experiment. Note that $\hat{\varvec{\tau }}_{ext}$ is external torque estimate. When detecting a collision, every element of $\hat{\varvec{\tau }}_{ext}$ should close to zero if the collision does not occur. However, τ_ext is affected by the payload, as shown in Fig. 7a. The reason for these results is that both τ_c and τ_p contribute to τ_ext in (2a). Therefore, it is difficult to detect a collision based on whether a certain element of $\hat{\varvec{\tau }}_{ext}$ exceeds the threshold. On the contrary, in Fig. 7b, the fourth element of CDI d_p exceeds the threshold d_th,p, which was set to 2 Nm for a collision, and d_p is not affected by the change in the payload, because it is independent of τ_p, as shown in (14). Therefore, the proposed collision detection method can detect collisions even in situations where the robot arm handles the unknown payload.

Fig. 7

Experimental results: a external torque estimate $\hat{\varvec{\tau }}_{ext}$ at joint 4, and b the fourth element of the CDI d_p

The CDI for contact tasks was verified in the experiments associated with impedance control for hand guiding, which is a typical contact task. Figure 8 shows the procedure of this experiment. Figure 8a shows the manipulator conducting impedance control, and the end-effector maintaining its initial position. An intended contact force F_e for hand guiding was applied to the end-effector in step 1 as shown in Fig. 8b. In step 2, a human–robot collision occurred as shown in Fig. 8c. In this step, an intended contact force F_e and the collision force F_c were applied to the manipulator simultaneously.

Fig. 8

Experimental condition for collision detection: a initial conditions, b intended contact between a human and manipulator, and c collisions with intended contact

Figure 9a, b show the values of $\hat{\varvec{\tau }}_{ext}$ at joint 2, which is mainly affected by F_e, and the second element of d_e during the experiment. Similar to the results in Fig. 7, Fig. 9 shows that τ_ext is affected not only by τ_c generated by F_c, but also by τ_e generated by F_e. However, elements of d_e were close to zero when only F_e was applied to the end-effector with no collision because d_e is independent of τ_e; however, the second element of d_e exceeded d_th,e when F_c was applied to the robot owing to collision. Therefore, any collisions that occur on the body of the robot while it conducts the contact task at the end-effector can be detected using the proposed collision detection method.

Fig. 9

Experimental results: a external torque estimate $\hat{\varvec{\tau }}_{ext}$ at joint 2, and the second element of the CDI d_e

6 Conclusions

In this study, a collision detection index was proposed to detect collisions of a robot arm when the robot performs a task involving a physical human–environment interaction. The proposed collision detection index was verified through experiments, and the following conclusions are drawn from the results:

• 1. The proposed scheme can be used to detect a collision, even in situations where the contact force between the manipulator and the object is unknown or uncertain. Therefore, it can be implemented in various collaborative tasks of a robot arm.

• 2. Using the proposed CDIs, the robot can detect collisions regardless of the frequency and magnitude of the collision force. Moreover, the proposed CDIs can be computed without any training data for machine learning. Therefore, the proposed collision detection method is a practical solution to ensure collision safety of collaborative robots.

One drawback of the proposed CDI is that its performance around the singular configurations of a robot arm can be degraded since the CDI is calculated based on the Jacobian matrix. However, such a situation can be avoided because robot arms generally do not operate around the singular configurations to prevent deterioration of control performance.

Accurate estimation of joint torque based on the motor current and friction estimate will be developed as a future study so that a CDI can be computed without any extra sensors.