1. Introduction

In recent years, manufacturing methods have shifted from low-mix, high-volume production to high-mix, low-volume production in response to consumer trends. In high-mix, low-volume production or variable-mix, variable-volume production, frequent changes in the production line make it difficult to spend considerable time teaching robots. Therefore, collaborative robots, which can be taught in a very short time using direct teaching methods and do not require safety fences unlike industrial robots, have begun to be used in industrial fields. Although collaborative robots offer advantages, such as space-saving operation and ease of teaching, safety measures are required to ensure human protection.

The collision-processing model proposed by Haddadin et al. classifies the process into seven categories: pre-collision, detection, isolation, identification, classification, reaction, and post-collision ¹. As sensorless collision-detection methods that do not require additional sensors other than joint position and current sensors, a collision-detection method using the difference between measured torque or current values and a dynamic model was proposed in 1989 ^2,3,4,5,6; subsequently, collision-detection methods based on changes in momentum owing to collisions were proposed ^7,8,9,10,11, and collision-detection methods based on changes in power owing to collisions were proposed ^12,13. A collision-detection method using deep learning was proposed in 2019 ^14,15, and a collision-detection method using support vector machines and autoencoders, which are machine learning techniques that do not require teaching on collision data, was proposed in 2022 ¹⁶. Other methods have been proposed, such as collision detection based on changes in vibration modes by analyzing arm vibrations ¹⁷ and collision detection using particle filters ¹².

Momentum-based methods have the advantage of not only detecting collisions but also identifying the link where a collision occurs. However, it is not possible to determine whether unknown manipulation forces during operation are applied to the end effector or whether collisions occur at the terminal link. Additionally, as time integration is used, a delay in collision detection may occur. Machine-learning-based methods have the advantage of not requiring a dynamic model or dynamic parameters and are robust to unknown manipulation forces during operation. However, they inherently suffer from the need to collect large amounts of training data, and if the training is inadequate, sufficient collision-detection performance cannot be achieved.

This paper proposes a sensorless collision-detection method that exploits static redundancy to achieve robustness against unknown manipulation forces and low-latency detection. This method requires no additional sensors and utilizes only the joint position and motor current sensors, which are commonly available in industrial and collaborative robots. This enables the detection of collisions at the terminal link under unknown manipulation force conditions by separating the joint driving torque into components generated exclusively by collisions and other components.

2. Collision-Detection Method

2.1. Formulation

The manipulator was an \(n\)-degree-of-freedom (\(n\)-DOF) serial-link manipulator that operated in an \(o\)-dimensional space with an \(m\)-dimensional workspace. The dimensional relationship satisfies either \(n \geq o > m\) or \(n > o \geq m\). The relationship between the manipulation force and joint driving torque of the manipulator can be expressed by the following static equation:

\begin{equation} \boldsymbol{\tau}_{r} = \boldsymbol{J}^{T} {}^{r}\!\boldsymbol{f}, \label{eq:Tau_rU003DJtFr} \end{equation}

where \(\boldsymbol{\tau}_{r} \in \mathbb{R}^{n}\) is the joint driving torque vector that generates the manipulation force \({}^{r}\!\boldsymbol{f}\); \({}^{r}\!\boldsymbol{f} \in \mathbb{R}^{m}\) is the manipulation force vector representing the force and moment applied in the workspace; and \(\boldsymbol{J} \in \mathbb{R}^{m \times n}\) is the Jacobian matrix, which is a function of the joint position vector \(\boldsymbol{q} \in \mathbb{R}^{n}\).

The manipulator has static redundancy because \(\boldsymbol{\tau}_r\) is an \(n\)-dimensional vector, and \({}^{r}\!\boldsymbol{f}\) is an \(m\)-dimensional vector. Therefore, Eq. \(\eqref{eq:Tau_rU003DJtFr}\) can be transformed into the following equation ¹⁸.

\begin{equation} \boldsymbol{\tau} = \boldsymbol{J}^{T} {}^{r}\!\boldsymbol{f} + \boldsymbol{U}\boldsymbol{s}, \label{eq:TauU003DJtFrU002BUs} \end{equation}

where \(\boldsymbol{\tau} \in \mathbb{R}^{n}\) is the joint driving torque vector, \(\boldsymbol{U}=[\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \dots, \boldsymbol{u}_{n-m}] \in \mathbb{R}^{n \times (n-m)}\) is an orthogonal matrix whose columns span the null-space of the Jacobian matrix \(\boldsymbol{J}\), and \(\boldsymbol{s} \in \mathbb{R}^{n-m}\) is a vector in the column space of matrix \(\boldsymbol{U}\). The relationship between matrices \(\boldsymbol{J}\) and \(\boldsymbol{U}\) is given by the following equation.

\begin{equation} \begin{bmatrix} \boldsymbol{J}^{T}& \boldsymbol{U} \end{bmatrix}^{-1} = \begin{bmatrix} \boldsymbol{J}^{+}& \boldsymbol{U} \end{bmatrix}^{T} \label{eq:U005BJtUU005DiU003DU005BJpUU005Dt} \end{equation}

As shown in Fig. 1, Eq. \(\eqref{eq:TauU003DJtFrU002BUs}\) represents the joint driving torque space. In the joint driving torque space, the row space of the Jacobian matrix \(\boldsymbol{J} = [\boldsymbol{j}_{1}, \boldsymbol{j}_{2}, \dots, \boldsymbol{j}_{m}]^{T}\) is spanned by the row vectors \(\boldsymbol{j}_{i} \in \mathbb{R}^{n}\). The joint driving torque vector \(\boldsymbol{J}^{T} {}^{r}\!\boldsymbol{f}\), which generates the manipulation force \({}^{r}\!\boldsymbol{f}\), lies within this row space. As the row space of the Jacobian matrix \(\boldsymbol{J}\) is an \(m\)-dimensional subspace of the joint driving torque space, there exists an \((n-m)\)-dimensional orthogonal complement, whose basis vectors are denoted by the vectors \(\boldsymbol{u}_{i}\).

As the manipulator performs the manipulation using its end effector, the manipulation forces \({}^{r}\!\boldsymbol{f}\) arising from the task are applied in the task space and are generally unknown. Industrial manipulators and collaborative robots are typically equipped with joint positions and motor current sensors on each joint. The motor current is proportional to the joint driving torque, which can be accessed by external systems. Each joint of a manipulator is typically controlled by a joint position controller, and the joint position follows a given reference. Therefore, when collision forces are applied owing to contact, the manipulator may either stop owing to overload protection or continue its motion, depending on the response of the system.

As the manipulator operates under joint position control, when external forces, such as manipulation or collision forces, are applied, the manipulator is not displaced by the external forces. Instead, joint driving torques are generated to counteract the external forces, resulting in equilibrium. Therefore, the following static equation holds between the joint driving torques of the manipulator and external forces, such as the manipulation and collision forces.

\begin{equation} \boldsymbol{\tau} = \boldsymbol{J}^{T} {}^{r}\!\boldsymbol{f} + \boldsymbol{J}_{c}^{T} {}^{c}\!\boldsymbol{f} = \boldsymbol{J}^{T} {}^{r}\!\boldsymbol{f} + \boldsymbol{\tau}_{t} + \boldsymbol{\tau}_{n}, \label{eq:TauU003DJtFr-JctFc} \end{equation}

where \(\boldsymbol{J}_{c} \in \mathbb{R}^{o \times n}\) is the Jacobian matrix at the collision point, and \({}^{c}\!\boldsymbol{f} \in \mathbb{R}^{o}\) is the collision force vector representing the force and moment applied at the collision point. As shown in Fig. 1, \(\boldsymbol{J}_{c}^{T} {}^{c}\!\boldsymbol{f}\) can be decomposed into a torque component \(\boldsymbol{\tau}_{t}\) in the row space of the Jacobian matrix \(\boldsymbol{J}\) and a component \(\boldsymbol{\tau}_{n}\) in the orthogonal complement. Let the torque component of the collision force projected onto the row space of the Jacobian matrix \(\boldsymbol{J}\) be defined as \(\boldsymbol{\tau}_{t} = \boldsymbol{J}^{T} {}^{t}\!\boldsymbol{f}\) and the net external forces projected onto the same space be defined as \(\boldsymbol{f} = {}^{r}\!\boldsymbol{f} + {}^{t}\!\boldsymbol{f}\). The following equation was obtained:

\begin{equation} \boldsymbol{\tau} = \boldsymbol{J}^{T} \boldsymbol{f} + \boldsymbol{\tau}_{n} . \label{eq:TauU003DJtFU002BTaun} \end{equation}

Furthermore, by defining the torque component of the collision force projected onto the orthogonal complement as \(\boldsymbol{\tau}_{n} = \boldsymbol{U} {}^{n}\!\boldsymbol{s}\), the following equation holds:

\begin{align} \boldsymbol{\tau} &= \begin{bmatrix} \boldsymbol{J}^{T}& \boldsymbol{U} \end{bmatrix} \begin{bmatrix} \boldsymbol{f} \\ {}^{n}\!\boldsymbol{s} \end{bmatrix} , \label{eq:eq6} \\ \end{align}

\begin{align} \begin{bmatrix} \boldsymbol{f} \\ {}^{n}\!\boldsymbol{s} \end{bmatrix} &= \begin{bmatrix} \bigl(\boldsymbol{J}^{+}\bigr)^{T} \\[3pt] \boldsymbol{U}^{T} \end{bmatrix} \boldsymbol{\tau} . \label{eq:FrtU003DJptTau} \end{align}

Substituting the external force \(\boldsymbol{f}\) projected onto the row space of the Jacobian matrix \(\boldsymbol{J}\) obtained from Eq. \(\eqref{eq:FrtU003DJptTau}\) into Eq. \(\eqref{eq:TauU003DJtFU002BTaun}\), the following equation is obtained:

\begin{equation} \boldsymbol{\tau}_{n} = \Bigl(\boldsymbol{E} - \boldsymbol{J}^{T} \bigl(\boldsymbol{J}^{+}\bigr)^{T}\Bigr) \boldsymbol{\tau}, \label{eq:Tau_nU003DU0028E-JtJptU0029Tau} \end{equation}

where \(\boldsymbol{E}\) is the identity matrix. In this study, \(\boldsymbol{\tau}_{n}\) refers to the null-space torque. Eq. \(\eqref{eq:Tau_nU003DU0028E-JtJptU0029Tau}\) derives the component of the joint driving torque in the orthogonal complement that counteracts collision forces, based on the Jacobian matrix \(\boldsymbol{J}\), which is a function of the joint position, and the joint driving torque \(\boldsymbol{\tau}\) output by the joint position controller. As Eq. \(\eqref{eq:Tau_nU003DU0028E-JtJptU0029Tau}\) is independent of the manipulation forces \({}^{r}\!\boldsymbol{f}\) in the task space, the null-space torque can be derived even when unknown manipulation forces are applied to the manipulator. Furthermore, as the null-space torque \(\boldsymbol{\tau}_{n}\) is the projection of external forces applied outside the task space onto the orthogonal complement of the Jacobian matrix \(\boldsymbol{J}\), it arises in response to the collision forces. Therefore, a sensorless collision-detection method that is robust to unknown manipulation forces can be achieved by monitoring the norm of the null-space torque, as shown in the following equation:

\begin{equation} \bigl\|\boldsymbol{\tau}_{n}\bigr\| = \Bigl\| \Bigl(\boldsymbol{E} - \boldsymbol{J}^{T} \bigl(\boldsymbol{J}^{+}\bigr)^{T}\Bigr) \boldsymbol{\tau} \Bigr\| > \sigma, \label{eq:U007CTau_nU007CU003DU007CU0028E-JtJptU0029TauU007C} \end{equation}

where \(\sigma\) is the threshold for collision detection.

2.2. Collision Detection in Limited Task Space

In industrial manipulators, the task space of an end effector is typically defined as a six-dimensional space consisting of its position and orientation. However, the manipulation forces applied to the end effector during operation do not necessarily act within the same space. For example, in a screw-tightening task, if the influence of the end effector position and orientation errors is neglected, the manipulation forces applied to the end effector lie in a two-dimensional space defined by the screw and rotational axes. Therefore, by appropriately defining the task space according to a specific task, the collision-detection performance can be improved. However, deriving Jacobian matrix \(\boldsymbol{J}\) for each task is cumbersome. Therefore, a collision-detection equation limited to the corresponding task space was derived by specifying the direction vector along which the manipulation force was applied.

Let \(\boldsymbol{v}_{i} \in \mathbb{R}^{o}\) (\(i = 1, 2, \dots, m\)) denote the direction vectors along which manipulation forces are applied. The space spanned by these vectors is referred to as the limited task space. As shown in Fig. 2, the limited task space forms an oblique coordinate system. Therefore, tensor algebra is used to project the external force \(\boldsymbol{f}\) in the task space onto the limited task space. In tensor algebra, the component form of the external force \(\boldsymbol{f}\) in the task space is expressed using contravariant components, and the actual vector \(\boldsymbol{f}\) is represented as follows:

\begin{equation} \boldsymbol{f} = f^{1}\boldsymbol{e}_{1} + f^{2}\boldsymbol{e}_{2} + \dots + f^{o}\boldsymbol{e}_{o}, \label{eq:FU003DfU005Eie_i} \end{equation}

where \(\boldsymbol{e}_{i}\) (\(i = 1, 2, \dots, o\)) represents the basis vectors in the task space, and \(f^{i}\) (\(i = 1, 2, \dots, o\)) represents the contravariant components of the vector \(\boldsymbol{f}\). As \(\boldsymbol{e}_{i}\) forms an orthogonal coordinate system, the contravariant components \(f^{i}\) are equal to the covariant components \(f_{i}\) (\(i = 1, 2, \dots, o\)). The covariant \({}^{l}\!f_i\) and contravariant \({}^{l}\!f^i\) components of the projected vector \({}^{l}\!\boldsymbol{f}\), which is the projection of \(\boldsymbol{f}\) onto the limited task space spanned by the basis vectors \(\boldsymbol{v}_{i}\), are given as follows:

\begin{align} {}^{l}\!f_i &= \boldsymbol{f} \cdot \boldsymbol{v}_{i}, &&i = 1, 2, \dots, m, \label{eq:eq11} \\ \end{align}

\begin{align} {}^{l}\!f^i &= \sum_{j=1}^{m} g^{ij} {}^{l}\!f_j, &&i = 1, 2, \dots, m, \label{eq:lFiU003Dg_jilFi} \\ \end{align}

\begin{align} g_{ij} &= \boldsymbol{v}_{i} \cdot \boldsymbol{v}_{j}, &&i = 1, 2, \dots, m, \quad j = 1, 2, \dots, m, \end{align}

where \(g_{ij}\) is the \((i, j)\) element of the covariant metric matrix, and \(g^{ij}\) is the \((i, j)\) element of the contravariant metric matrix, which is the inverse of the covariant metric matrix. Therefore, the vector \({}^{l}\!\boldsymbol{f} = [{}^{l}\!f{\,}^{i}]\) projected onto the limited task space is given by

\begin{equation} {}^{l}\!\boldsymbol{f} = \boldsymbol{V}{\,}\boldsymbol{G}^{-1} \boldsymbol{V}^{{\,}T} \boldsymbol{f}, \label{eq:lFU003DVGiVtF} \end{equation}

where \(\boldsymbol{V} = [\boldsymbol{v}_{1}, \boldsymbol{v}_{2}, \dots, \boldsymbol{v}_{m}]\) is the matrix constructed from the basis vectors \(\boldsymbol{v}_i\) of the limited task space, \(\boldsymbol{G}^{-1} = [g^{ij}]\) is the contravariant metric matrix, and \(\boldsymbol{f} = [f^{1}, f^{2}, \dots, f^{o}]^{T}\) is the component form of the external force \(\boldsymbol{f}\) in the task space. Therefore, by substituting Eq. \(\eqref{eq:lFU003DVGiVtF}\) into the external force \(\boldsymbol{f}\) of Eq. \(\eqref{eq:TauU003DJtFU002BTaun}\), the collision-detection equation constrained to the limited task space is derived as follows:

\begin{equation} \bigl\| \boldsymbol{\tau}_{n} \bigr\| = \Bigl\| \Bigl(\boldsymbol{E} - \boldsymbol{J}^{{\,}T} \boldsymbol{V}{\,} \boldsymbol{G}^{-1} \boldsymbol{V}^{{\,}T} \bigl(\boldsymbol{J}^{{\,}+}\bigr)^{T} \Bigr) \boldsymbol{\tau} \Bigr\| > \sigma. \label{eq:U007CTau_nU007CU003DU007CU0028E-JtVGiVtJptU0029TauU007C} \end{equation}

Furthermore, if the basis vectors \(\boldsymbol{v}_{i}\) of the limited task space form an orthonormal coordinate system, the covariant metric matrix \(\boldsymbol{G}\) becomes an identity matrix, and the collision-detection equation is simplified as follows:

\begin{equation} \bigl\| \boldsymbol{\tau}_{n} \bigr\| = \Bigl\| \Bigl(\boldsymbol{E} - \boldsymbol{J}^{{\,}T} \boldsymbol{V}{\,} \boldsymbol{V}^{{\,}T} \bigl(\boldsymbol{J}^{{\,}+}\bigr)^{T} \Bigr) \boldsymbol{\tau} \Bigr\| > \sigma. \label{eq:eq15} \end{equation}

3. Simulation

In this simulation, the manipulation and collision forces were applied to a planar two-link manipulator to demonstrate that the proposed method is robust to unknown manipulation forces and can detect collisions at the terminal link, including the end effector, even outside the limited task space.

The manipulator used in the simulation was a planar two-link manipulator with link lengths \(l_1 = 0.2\) m and \(l_2 = 0.15\) m, as shown in Fig. 3. Each joint was assumed to be controlled by a joint position controller, such that the end effector position was maintained at \(\boldsymbol{r} = [0.3,{\ }0]^T\) m. It was assumed that the joint driving torques counteracted the external forces without causing any joint position errors, and the simulation was conducted under static conditions. The limited task space was defined by the direction vector \(\boldsymbol{v} = [1,{\ }1]^T\). A manipulation force \({}^{r}\!\boldsymbol{f} = [1,{\ }1]^T\) N was applied to the tool center point (TCP). Three collision forces were applied as follows: \({}^{c1}\!\boldsymbol{f} = [0,{\ }-1]^T\) N at the TCP and \({}^{c2}\!\boldsymbol{f} = [1,{\ }1]^T\) N and \({}^{c3}\!\boldsymbol{f} = [1,{\ }0.705]^T\) N at a point located 0.075 m from the second joint on the second link. Table 1 shows the joint torque \(\boldsymbol{\tau}\), the norm of the null-space torque \(\|\boldsymbol{\tau}_{n}\|\), and the collision-detection results with a threshold of \(\sigma = 10^{-5}\) for each applied external force, and the case where both the manipulation force \({}^{r}\!\boldsymbol{f}\) and the collision force \({}^{c1}\!\boldsymbol{f}\) are applied simultaneously.

According to Table 1, the norm of the null-space torque differs by approximately \(10^{15}\) between the case in which only the manipulation force \({}^{r}\!\boldsymbol{f}\) is applied and the cases in which the collision forces \({}^{c1}\!\boldsymbol{f}\) or \({}^{c2}\!\boldsymbol{f}\) are applied. A similar result was observed when both the manipulation force \({}^{r}\!\boldsymbol{f}\) and the collision force \({}^{c1}\!\boldsymbol{f}\) were applied, indicating that the proposed method achieved robust collision detection against manipulation forces. Moreover, although the collision force \({}^{c2}\!\boldsymbol{f}\) was applied to the terminal link in a direction parallel to the limited task space, it was successfully detected. Similarly, although the collision force \({}^{c1}\!\boldsymbol{f}\) is applied at the same TCP location as the manipulation force and in a direction different from that of the limited task space, the collision was successfully detected. However, when the collision force \({}^{c3}\!\boldsymbol{f}\) was applied, no collision was detected, although it differed only slightly in the \(y\)-axis component from the collision force \({}^{c2}\!\boldsymbol{f}\).

4. Actual Experiment

In actual experiments, collision detection is performed by applying manipulation and collision forces to the terminal link of a vertical serial-link industrial manipulator. By incorporating gravity and friction torque compensation, we demonstrated that the proposed method is applicable to real hardware.

An xArm 6 manipulator (UFactory) was used for the experiments. The main specifications of xArm 6 are listed in Table 2. xArm 6 is controlled using the ROS by sending linear motion commands to the TCP in Cartesian coordinates while the joint motor current values are measured. The joint driving torque can be obtained from the motor current values using torque constants. Therefore, following the method proposed by Gaz et al. ¹⁹, the joint driving torque counteracting the external forces was estimated by identifying and compensating for the torque constant, gravity torque parameters, and dynamic friction coefficients through preliminary experiments. In the first step, the dynamic friction coefficients were identified among the unknown parameters. The dynamic friction torque was modeled using a sigmoid function as follows:

According to Fig. 7, the norm of the null-space torque reaches its maximum during the transition from command No.1 to command No.2 with a value of approximately 9 Nm. This transition from command No.1 to No.2 corresponds to the moment when the largest collision force is applied, and the norm of the null-space torque increases accordingly. However, during the transition from command No.6 to No.5, where the manipulation force applied to the end effector reaches its maximum, the norm of the null-space torque is approximately 3.5 Nm, which is smaller than the value observed when a collision force is applied. However, during command Nos.5 and 4, the norm of the null-space torque remains at approximately 6 Nm, despite the absence of external forces, indicating a relatively high value. This is likely because the gravity and dynamic friction torques cannot be fully compensated by the respective compensation methods. This was attributed to identification errors in the gravity torque parameters and dynamic friction coefficients obtained using the least-squares method. According to Fig. 8, it can be observed that, except around the transition from command No.1 to No.2 and from command No.6 to No.5 where external forces are applied to the arm, the joint driving torques counteracting external forces are small. However, residual joint driving torques of approximately 4 Nm remain, indicating that the compensation for the gravity and dynamic friction torques is imperfect. In actual production, false collision detection in the absence of real collisions reduces the productivity. Therefore, while setting conservative thresholds, it is important to avoid false detections; excessively high thresholds suppress false alarms but fail to detect minor collisions. The purpose of collision detection is to ensure the safety of collaboration with humans and to prevent damage to the robot itself and the environment. Therefore, the speed of collision detection is critical to prevent severe accidents, whereas the accuracy of collision detection using only minor collision forces is of relatively low importance. Therefore, in the experiment validation shown in Fig. 7, the collision-detection threshold was set to 7.2 Nm, corresponding to 1.2 times the maximum null-space torque norm during operations without external collision forces. This threshold suppresses false detections while preventing serious accidents, thereby enabling reliable collision detection with minimal false detections even under conditions with an unknown manipulation force.

5. Discussion

This paper proposes a sensorless collision-detection method that exploits static redundancy to achieve robustness against unknown manipulation forces and low-latency detection.

Conventional methods, such as those that directly monitor motor currents ^2,3,4,5,6, and momentum-observer-based methods ^7,8,9,10,11, detect collisions by comparing the joint driving torque generated in response to external collision forces with the joint torque estimated from a manipulator model. When a collision occurs, the joint torque from the base to the link, where the collision force is applied, deviates from the manipulator model, thereby enabling collision detection. However, when a collision occurs at a terminal link, the resulting deviation from the model cannot be distinguished from that caused by unknown manipulation forces, making collision detection infeasible under such conditions. The method using the power index proposed in ^12,13 addresses this issue by enabling the detection of the collision location, thereby allowing discrimination between unknown manipulation forces and collision forces applied to the terminal link. Consequently, collision detection becomes feasible even under unknown manipulation force conditions. However, as time integration is used, a detection delay of approximately 40 ms occurs after the collision. Methods based on deep learning, autoencoders, and support vector machines ^14,15,16 can detect collisions at the terminal link even under unknown manipulation force conditions, achieving a detection delay as low as 10 ms. However, as these methods are based on machine learning, they require a large amount of training data, and insufficient training can result in poor collision-detection performance. In contrast, the method proposed in this paper exploits static redundancy to separate the joint driving torque in a limited task space from its null-space component for collision detection. This allows the detection of collisions at the terminal link even under unknown manipulation force conditions. Moreover, as the method only monitors the joint torque components within the subspaces, the detection speed is comparable to that of conventional methods based on the direct monitoring of joint torques. In addition, as the norm of the null-space torque is computed using the Jacobian matrix of the end effector, the proposed method is computationally simple. However, as the proposed method is based on static redundancy for collision detection, it cannot be applied to tasks in which static redundancy does not exist.

In the simulation results shown in Table 1, the application of the collision force \({}^{c3}\!\boldsymbol{f}\) is labeled as “No Collision,” although a collision force is applied. This detection failure occurs because there exists a manipulation force that produces the same joint driving torque vector as that generated by the collision force \({}^{c3}\!\boldsymbol{f}\). As this manipulation force lies within a limited task space, it cannot be distinguished from manipulation forces. This phenomenon, in which different external forces produce the same torque, is analogous to the force-moment relationship, where different application points of external forces can produce the same moment. As shown in Fig. 9, when represented in the joint torque space, the joint driving torque vector \(\boldsymbol{J}_{c}^{{\,}T} {}^{c3}\!\boldsymbol{f}\) counteracting the collision force lies on the hyperplane spanned by the Jacobian \(\boldsymbol{J}_{c}^{{\,}T}\) associated with the collision point. This torque vector is decomposed into \(\boldsymbol{\tau}_t\), which lies on the hyperplane \(\boldsymbol{J}^{{\,}T} \boldsymbol{V}{\,}\boldsymbol{G}^{-1} \boldsymbol{V}^{{\,}T}\) corresponding to the limited task space, and \(\boldsymbol{\tau}_n\), which lies on the orthogonal complement \(\boldsymbol{u}_1\). However, there are cases in which the hyperplane spanned by the Jacobian \(\boldsymbol{J}_{c}^{{\,}T}\) associated with the collision point intersects with the hyperplane \(\boldsymbol{J}^{{\,}T} \boldsymbol{V}{\,}\boldsymbol{G}^{-1} \boldsymbol{V}^{{\,}T}\) corresponding to the limited task space. Therefore, when the joint driving torque vector \(\boldsymbol{J}_{c}^{{\,}T} {}^{c3}\!\boldsymbol{f}\) counteracting the collision force lies in the intersection subspace of the two hyperplanes, the distance from the limited task space (i.e., the norm of the null-space torque vector) becomes zero, resulting in collision-detection failure. In principle, these collisions cannot be detected using the proposed method. Therefore, the proposed collision-detection method must be combined with other detection methods to compensate for cases in which collisions cannot be detected using this approach alone.

In the experimental results shown in Fig. 7, the norm of the null-space torque reaches a relatively high value of approximately 5 Nm during motion toward command No.4, where no external force is applied after the end effector separates from the force gauge following the manipulation phase. In the experiments presented in this paper, gravity and friction torque compensation was performed using parameters identified by the least-squares method to estimate the joint driving torque that counteracts the external forces applied to the manipulator. As shown in Fig. 8, the compensated joint driving torque exhibits high values during the transitions from command No.6 to No.5 and command No.1 to No.2 in the actual experimental motion. This is attributed to the reversal of the motion direction of the end effector, which results in a static friction condition. Furthermore, at approximately 25 s, where the null-space torque exhibits a high value despite no external forces being applied in Fig. 7, the compensated joint driving torque at the third joint, which counteracts the external forces, as shown in Fig. 8, also exhibits a relatively high value. These observations suggest that the performance of the proposed collision-detection method is highly dependent on the accuracy of the gravity and friction torque compensation.

6. Conclusions

This paper proposed a sensorless collision-detection method that exploits static redundancy, achieving robustness against unknown manipulation forces and a short detection delay. In addition, simulations using a planar 2-DOF manipulator and actual experiments using a vertical 6-DOF manipulator were conducted, and the following results were obtained:

• Collisions, including those at the terminal link, could be detected even under unknown manipulation force conditions.

• By appropriately limiting the task space according to the task, static redundancy can be introduced, enabling the application of the proposed collision-detection method.

• In principle, collisions in which the joint torque counteracting the collision force lies within the row space of the Jacobian matrix cannot be detected using the proposed method. Therefore, it is necessary to compensate for such cases by combining this method with other collision-detection methods.

In future studies, we will combine the proposed method with other approaches to detect collisions whose counteracting torques lie within the row space of the Jacobian matrix. In addition, we plan to investigate the dynamic characteristics and extend the method to allow reliable collision detection even during high-speed motion.