8 research outputs found

    Feasibility Investigation of Obstacle-Avoiding Sensors Unit without Image Processing

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    Feasibility of a simple method to detect step height, slope angle, and trench width using four infrared-light-source PSD range sensors is examined, and the reproducibility and accuracy of characteristic parameter detection are also examined. Detection error of upward slope angle is within 2.5 degrees, while it is shown that the detection error of downward slope angle exceeding 20 degrees is very large. In order to reduce such errors, a method to improve range-voltage performance of a range sensor is proposed, and its availability is demonstrated. We also show that increase in trial frequency is a better way, although so as not to increase the detection delay. Step height is identified with an error of ยฑ1.5โ€‰mm. It is shown that trench width cannot be reliably measured at this time. It is suggested that an additional method is needed if we have to advance the field of obstacle detection

    Real-Time Simulation For Control Of Soft Robots With Self-Collisions Using Model Order Reduction For Contact Forces

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    International audienceIn rigid robotics, self-collision are usually avoided since it leads to a failure in the robot control and can also cause damage. In soft robotics, the situation is very different, and self-collisions may even be a desirable property, for example to gain artificial stiffness or to provide a natural limitation to the workspace. However, the modeling and simulation of self-collision is very costly as it requires first a collision detection algorithm to detect where collisions occur, and most importantly, it requires solving a constrained problem to avoid interpenetrations. When the number of contact points is large, this computation slows down the simulation dramatically. In this paper, we apply a numerical method to alleviate the contact response computation by reducing the contact space in a lowdimensional positive space obtained from experiments. We show good accuracy while speeding up dramatically the simulation. We apply the method in simulation on a cable-actuated finger and on a continuum manipulator performing exploration. We also show that the reduced contact method proposed can be used for inverse modeling. The method can therefore be used for control or design

    Active Collision Avoidance for Human-Robot Interaction With UKF, Expert System, and Artificial Potential Field Method

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    With the development of Industry 4.0, the cooperation between robots and people is increasing. Therefore, manโ€”machine security is the first problem that must be solved. In this paper, we proposed a novel methodology of active collision avoidance to safeguard the human who enters the robot's workspace. In the conventional approaches of obstacle avoidance, it is not easy for robots and humans to work safely in the common unstructured environment due to the lack of the intelligence. In this system, one Kinect is employed to monitor the workspace of the robot and detect anyone who enters the workspace of the robot. Once someone enters the working space, the human will be detected, and the skeleton of the human can be calculated in real time by the Kinect. The measurement errors increase over time, owing to the tracking error and the noise of the device. Therefore we use an Unscented Kalman Filter (UKF) to estimate the positions of the skeleton points. We employ an expert system to estimate the behavior of the human. Then let the robot avoid the human by taking different measures, such as stopping, bypassing the human or getting away. Finally, when the robot needs to execute bypassing the human in real time, to achieve this, we adopt a method called artificial potential field method to generate a new path for the robot. By using this active collision avoidance, the system can achieve the purpose that the robot is unable to touch on the human. This proposed system highlights the advantage that during the process, it can first detect the human, then analyze the motion of the human and finally safeguard the human. We experimentally tested the active collision avoidance system in real-world applications. The results of the test indicate that it can effectively ensure human security

    Integration of Reactive, Torque-Based Self-Collision Avoidance Into a Task Hierarchy

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    Reactively dealing with self-collisions is an important requirement on multi-DOF robots in unstructured and dynamic environments. Classical methods to integrate respective algorithms into task hierarchies cause substantial problems: Either these unilateral safety constraints are permanently active, unnecessarily locking DOF for other tasks, or they get activated online and result in a discontinuous control law. We propose a new, reactive self-collision avoidance algorithm for highly complex robotic systems with a large number of DOF. In particular, configuration dependent damping is imposed to dissipate undesired kinetic energy in a well-directed manner. Moreover, we merge the algorithm with a novel method to incorporate these unilateral constraints into a dynamic task hierarchy. Our approach both allows to specifically limit the force/torque derivative to comply with physical constraints of the real robot and to prevent discontinuities in the control law while activating/deactivating the constraints. No redundancy is wasted. No comparable algorithms have been developed and implemented on a torque controlled robot with such a level of complexity so far. The implementation of our generic solution on the multi-DOF humanoid Justin clearly validates the performance and demonstrates the real-time applicability of our synthetic approach. The proposed method can be used to contribute to whole-body controllers

    ์•ˆ์ „ํ•œ ์žฌ๊ตฌ์„ฑ ๋กœ๋ด‡ ์‹œ์Šคํ…œ: ์„ค๊ณ„, ํ”„๋กœ๊ทธ๋ž˜๋ฐ ๋ฐ ๋ฐ˜์‘ํ˜• ๊ฒฝ๋กœ๊ณ„ํš

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    ํ•™์œ„๋…ผ๋ฌธ (๋ฐ•์‚ฌ) -- ์„œ์šธ๋Œ€ํ•™๊ต ๋Œ€ํ•™์› : ๊ณต๊ณผ๋Œ€ํ•™ ๊ธฐ๊ณ„ํ•ญ๊ณต๊ณตํ•™๋ถ€, 2020. 8. ๋ฐ•์ข…์šฐ.The next generation of robots are being asked to work in close proximity to humans. At the same time, the robot should have the ability to change its topology to flexibly cope with various tasks. To satisfy these two requirements, we propose a novel modular reconfi gurable robot and accompanying software architecture, together with real-time motion planning algorithms to allow for safe operation in unstructured dynamic environments with humans. Two of the key innovations behind our modular manipulator design are a genderless connector and multi-dof modules. By making the modules connectable regardless of the input/output directions, a genderless connector increases the number of possible connections. The developed genderless connector can transmit as much load as necessary to an industrial robot. In designing two-dof modules, an offset between two joints is imposed to improve the overall integration and the safety of the modules. To cope with the complexity in modeling due to the genderless connector and multi-dof modules, a programming architecture for modular robots is proposed. The key feature of the proposed architecture is that it efficiently represents connections of multi-dof modules only with connections between modules, while existing architectures should explicitly represent all connections between links and joints. The data structure of the proposed architecture contains properties of tree-structured multi-dof modules with intra-module relations. Using the data structure and connection relations between modules, kinematic/dynamic parameters of connected modules can be obtained through forward recursion. For safe operation of modular robots, real-time robust collision avoidance algorithms for kinematic singularities are proposed. The main idea behind the algorithms is generating control inputs that increase the directional manipulability of the robot to the object direction by reducing directional safety measures. While existing directional safety measures show undesirable behaviors in the vicinity of the kinematic singularities, the proposed geometric safety measure generates stable control inputs in the entire joint space. By adding the preparatory input from the geometric safety measure to the repulsive input, a hierarchical collision avoidance algorithm that is robust to kinematic singularity is implemented. To mathematically guarantee the safety of the robot, another collision avoidance algorithm using the invariance control framework with velocity-dependent safety constraints is proposed. When the object approached the robot from a singular direction, the safety constraints are not satis ed in the initial state of the robot and the safety cannot be guaranteed using the invariance control. By proposing a control algorithm that quickly decreases the preparatory constraints below thresholds, the robot re-enters the constraint set and avoids collisions using the invariance control framework. The modularity and safety of the developed reconfi gurable robot is validated using a set of simulations and hardware experiments. The kinematic/dynamic model of the assembled robot is obtained in real-time and used to accurately control the robot. Due to the safe design of modules with o sets and the high-level safety functions with collision avoidance algorithms, the developed recon figurable robot has a broader safe workspace and wider ranger of safe operation speed than those of cooperative robots.๋‹ค์Œ ์„ธ๋Œ€์˜ ๋กœ๋ด‡์€ ์‚ฌ๋žŒ๊ณผ ๊ฐ€๊นŒ์ด์—์„œ ํ˜‘์—…ํ•  ์ˆ˜ ์žˆ๋Š” ๊ธฐ๋Šฅ์„ ๊ฐ€์ ธ์•ผํ•œ๋‹ค. ๊ทธ์™€ ๋™์‹œ์—, ๋กœ๋ด‡์€ ๋‹ค์–‘ํ•˜๊ฒŒ ๋ณ€ํ•˜๋Š” ์ž‘์—…์— ๋Œ€ํ•ด ์œ ์—ฐํ•˜๊ฒŒ ๋Œ€์ฒ˜ํ•  ์ˆ˜ ์žˆ๋„๋ก ์ž์‹ ์˜ ๊ตฌ์กฐ๋ฅผ ๋ฐ”๊พธ๋Š” ๊ธฐ๋Šฅ์„ ๊ฐ€์ ธ์•ผ ํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ๋‘ ๊ฐ€์ง€ ์š”๊ตฌ์กฐ๊ฑด์„ ๋งŒ์กฑ์‹œํ‚ค๊ธฐ ์œ„ํ•ด, ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ์ƒˆ๋กœ์šด ๋ชจ๋“ˆ๋ผ ๋กœ๋ด‡ ์‹œ์Šคํ…œ๊ณผ ํ”„๋กœ๊ทธ๋ž˜๋ฐ ์•„ํ‚คํ…์ณ๋ฅผ ์ œ์‹œํ•˜๊ณ , ์‚ฌ๋žŒ์ด ์กด์žฌํ•˜๋Š” ๋™์  ํ™˜๊ฒฝ์—์„œ ์•ˆ์ „ํ•œ ๋กœ๋ด‡์˜ ์šด์šฉ์„ ์œ„ํ•œ ์‹ค์‹œํ•œ ๊ฒฝ๋กœ ๊ณ„ํš ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์‹œํ•œ๋‹ค. ๊ฐœ๋ฐœ๋œ ๋ชจ๋“ˆ๋ผ ๋กœ๋ด‡์˜ ๋‘ ๊ฐ€์ง€ ํ•ต์‹ฌ์ ์ธ ํ˜์‹ ์„ฑ์€ ๋ฌด์„ฑ๋ณ„ ์ปค๋„ฅํ„ฐ์™€ ๋‹ค์ž์œ ๋„ ๋ชจ๋“ˆ์—์„œ ์ฐพ์„ ์ˆ˜ ์žˆ๋‹ค. ์ž…๋ ฅ/์ถœ๋ ฅ ๋ฐฉํ–ฅ์— ์ƒ๊ด€ ์—†์ด ๋ชจ๋“ˆ์ด ์—ฐ๊ฒฐ๋  ์ˆ˜ ์žˆ๋„๋ก ํ•จ์œผ๋กœ์จ, ๋ฌด์„ฑ๋ณ„ ์ปค๋„ฅํ„ฐ๋Š” ๊ฒฐํ•ฉ ๊ฐ€๋Šฅํ•œ ๊ฒฝ์šฐ์˜ ์ˆ˜๋ฅผ ๋Š˜๋ฆด ์ˆ˜ ์žˆ๋‹ค. ๊ฐœ๋ฐœ๋œ ๋ฌด์„ฑ๋ณ„ ์ปค๋„ฅํ„ฐ๋Š” ์‚ฐ์—…์šฉ ๋กœ๋ด‡์—์„œ ์š”๊ตฌ๋˜๋Š” ์ถฉ๋ถ„ํ•œ ๋ถ€ํ•˜๋ฅผ ๊ฒฌ๋”œ ์ˆ˜ ์žˆ๋„๋ก ์„ค๊ณ„๋˜์—ˆ๋‹ค. 2 ์ž์œ ๋„ ๋ชจ๋“ˆ์˜ ์„ค๊ณ„์—์„œ ๋‘ ์ถ• ์‚ฌ์ด์— ์˜คํ”„์…‹์„ ๊ฐ€์ง€๋„๋ก ํ•จ์œผ๋กœ์จ ์ „์ฒด์ ์ธ ์™„์„ฑ๋„ ๋ฐ ์•ˆ์ „๋„๋ฅผ ์ฆ๊ฐ€์‹œ์ผฐ๋‹ค. ๋ฌด์„ฑ๋ณ„ ์ปค๋„ฅํ„ฐ์™€ ๋‹ค์ž์œ ๋„ ๋ชจ๋“ˆ๋กœ ์ธํ•œ ๋ชจ๋ธ๋ง์˜ ๋ณต์žก์„ฑ์— ๋Œ€์‘ํ•˜๊ธฐ ์œ„ํ•ด, ์ผ๋ฐ˜์ ์ธ ๋ชจ๋“ˆ๋ผ ๋กœ๋ด‡์„ ์œ„ํ•œ ์†Œํ”„ํŠธ์›จ์–ด ์•„ํ‚คํ…์ณ๋ฅผ ์ œ์•ˆํ•˜์˜€๋‹ค. ๊ธฐ์กด ๋ชจ๋“ˆ๋ผ ๋กœ๋ด‡์˜ ์—ฐ๊ฒฐ์„ ๋‚˜ํƒ€๋‚ด๋Š” ๋ฐฉ๋ฒ•์ด ๋ชจ๋“  ๋งํฌ์™€ ์กฐ์ธํŠธ ์‚ฌ์ด์˜ ์—ฐ๊ฒฐ ๊ด€๊ณ„๋ฅผ ๋ณ„๋„๋กœ ๋‚˜ํƒ€๋‚ด์•ผํ•˜๋Š” ๊ฒƒ๊ณผ ๋‹ค๋ฅด๊ฒŒ, ์ œ์•ˆ๋œ ์•„ํ‚คํ…์ณ๋Š” ๋ชจ๋“ˆ๋“ค ์‚ฌ์ด์˜ ์—ฐ๊ฒฐ๊ด€๊ณ„๋งŒ์„ ๋‚˜ํƒ€๋ƒ„์œผ๋กœ์จ ํšจ์œจ์ ์ธ ๋‹ค์ž์œ ๋„ ๋ชจ๋“ˆ์˜ ์—ฐ๊ฒฐ๊ด€๊ณ„๋ฅผ ๋‚˜ํƒ€๋‚ผ ์ˆ˜ ์žˆ๋‹ค๋Š” ๊ฒƒ์„ ํŠน์ง•์œผ๋กœ ํ•œ๋‹ค. ์ด๋ฅผ ์œ„ํ•ด ํŠธ๋ฆฌ ๊ตฌ์กฐ๋ฅผ ๊ฐ€์ง€๋Š” ์ผ๋ฐ˜์ ์ธ ๋‹ค์ž์œ ๋„ ๋ชจ๋“ˆ์˜ ์„ฑ์งˆ์„ ๋‚˜ํƒ€๋‚ด๋Š” ๋ฐ์ดํ„ฐ ๊ตฌ์กฐ๋ฅผ ์ •์˜ํ•˜์˜€๋‹ค. ๋ชจ๋“ˆ๋“ค ์‚ฌ์ด์˜ ์—ฐ๊ฒฐ๊ด€๊ณ„ ๋ฐ ๋ฐ์ดํ„ฐ ๊ตฌ์กฐ๋ฅผ ์ด์šฉํ•˜์—ฌ, ์ •ํ™•ํ•œ ๊ธฐ๊ตฌํ•™/๋™์—ญํ•™ ๋ชจ๋ธ ํŒŒ๋ผ๋ฏธํ„ฐ๋ฅผ ์–ป์–ด๋‚ด๋Š” ์ˆœ๋ฐฉํ–ฅ ์žฌ๊ท€ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ๊ตฌํ˜„ํ•˜์˜€๋‹ค. ๋ชจ๋“ˆ๋ผ ๋กœ๋ด‡์˜ ์•ˆ์ „ํ•œ ์šด์šฉ์„ ์œ„ํ•ด, ๊ธฐ๊ตฌํ•™์  ํŠน์ด์ ์— ๊ฐ•๊ฑดํ•œ ์‹ค์‹œ๊ฐ„ ์ถฉ๋ŒํšŒํ”ผ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์•ˆํ•˜์˜€๋‹ค. ๋ฐฉํ–ฅ์„ฑ ์•ˆ์ „๋„๋ฅผ ์ค„์ด๋Š” ๋ฐฉํ–ฅ์˜ ์ œ์–ด ์ž…๋ ฅ์„ ์ƒ์„ฑํ•˜์—ฌ ๋ฌผ์ฒด ๋ฐฉํ–ฅ์œผ๋กœ์˜ ๋กœ๋ด‡ ๋ฐฉํ–ฅ์„ฑ ๋งค๋‹ˆํ“ฐ๋Ÿฌ๋นŒ๋ฆฌํ‹ฐ๋ฅผ ์ฆ๊ฐ€์‹œํ‚ค๋Š” ๊ฒƒ์ด ์ œ์•ˆํ•œ ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ํ•ต์‹ฌ์ด๋‹ค. ๊ธฐ์กด์˜ ๋ฐฉํ–ฅ์„ฑ ์•ˆ์ „๋„๊ฐ€ ๊ธฐ๊ตฌํ•™์  ํŠน์ด์  ๊ทผ์ฒ˜์—์„œ ์›ํ•˜์ง€ ์•Š๋Š” ์„ฑ์งˆ์„ ๊ฐ€์ง€๋Š” ๊ฒƒ๊ณผ๋Š” ๋ฐ˜๋Œ€๋กœ, ์ œ์•ˆํ•œ ๊ธฐํ•˜ํ•™์  ์•ˆ์ „๋„๋Š” ์ „์ฒด ์กฐ์ธํŠธ ๊ณต๊ฐ„์—์„œ ์•ˆ์ •์ ์ธ ์ œ์–ด ์ž…๋ ฅ์„ ์ƒ์„ฑํ•œ๋‹ค. ์ด ๊ธฐํ•˜ํ•™์  ์•ˆ์ „๋„๋ฅผ ์ด์šฉํ•˜์—ฌ, ๊ธฐ๊ตฌํ•™์  ํŠน์ด์ ์— ๊ฐ•๊ฑดํ•œ ๊ณ„์ธต์  ์ถฉ๋ŒํšŒํ”ผ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ๊ตฌํ˜„ํ•˜์˜€๋‹ค. ์ˆ˜ํ•™์ ์œผ๋กœ ๋กœ๋ด‡์˜ ์•ˆ์ „๋„๋ฅผ ๋ณด์žฅํ•˜๊ธฐ ์œ„ํ•ด, ์ƒ๋Œ€์†๋„์— ์ข…์†์ ์ธ ์•ˆ์ „ ์ œ์•ฝ์กฐ๊ฑด์„ ๊ฐ€์ง€๋Š” ๋ถˆ๋ณ€ ์ œ์–ด ํ”„๋ ˆ์ž„์›Œํฌ์„ ์ด์šฉํ•˜์—ฌ ๋˜ ํ•˜๋‚˜์˜ ์ถฉ๋Œ ํšŒํ”ผ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์•ˆํ•˜์˜€๋‹ค. ๋ฌผ์ฒด๊ฐ€ ํŠน์ด์  ๋ฐฉํ–ฅ์œผ๋กœ๋ถ€ํ„ฐ ๋กœ๋ด‡์— ์ ‘๊ทผํ•  ๋•Œ, ๋กœ๋ด‡์˜ ์ดˆ๊ธฐ ์ƒํƒœ์—์„œ ์•ˆ์ „ ์ œ์•ฝ์กฐ๊ฑด์„ ๋งŒ์กฑ์‹œํ‚ค์ง€ ๋ชปํ•˜๊ฒŒ ๋˜์–ด ๋ถˆ๋ณ€์ œ์–ด๋ฅผ ์ ์šฉํ•  ์ˆ˜ ์—†๊ฒŒ ๋œ๋‹ค. ์ค€๋น„ ์ œ์•ฝ์กฐ๊ฑด์„ ๋น ๋ฅด๊ฒŒ ์ž„๊ณ„์  ์•„๋ž˜๋กœ ๊ฐ์†Œ์‹œํ‚ค๋Š” ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ ์šฉํ•จ์œผ๋กœ์จ, ๋กœ๋ด‡์€ ์ œ์•ฝ์กฐ๊ฑด ์ง‘ํ•ฉ์— ๋‹ค์‹œ ๋“ค์–ด๊ฐ€๊ณ  ๋ถˆ๋ณ€ ์ œ์–ด ๋ฐฉ๋ฒ•์„ ์ด์šฉํ•˜์—ฌ ์ถฉ๋Œ์„ ํšŒํ”ผํ•  ์ˆ˜ ์žˆ๊ฒŒ ๋œ๋‹ค. ๊ฐœ๋ฐœ๋œ ์žฌ๊ตฌ์„ฑ ๋กœ๋ด‡์˜ ๋ชจ๋“ˆ๋ผ๋ฆฌํ‹ฐ์™€ ์•ˆ์ „๋„๋Š” ์ผ๋ จ์˜ ์‹œ๋ฎฌ๋ ˆ์ด์…˜๊ณผ ํ•˜๋“œ์›จ์–ด ์‹คํ—˜์„ ํ†ตํ•ด ๊ฒ€์ฆ๋˜์—ˆ๋‹ค. ์‹ค์‹œ๊ฐ„์œผ๋กœ ์กฐ๋ฆฝ๋œ ๋กœ๋ด‡์˜ ๊ธฐ๊ตฌํ•™/๋™์—ญํ•™ ๋ชจ๋ธ์„ ์–ป์–ด๋‚ด ์ •๋ฐ€ ์ œ์–ด์— ์‚ฌ์šฉํ•˜์˜€๋‹ค. ์•ˆ์ „ํ•œ ๋ชจ๋“ˆ ๋””์ž์ธ๊ณผ ์ถฉ๋Œ ํšŒํ”ผ ๋“ฑ์˜ ๊ณ ์ฐจ์› ์•ˆ์ „ ๊ธฐ๋Šฅ์„ ํ†ตํ•˜์—ฌ, ๊ฐœ๋ฐœ๋œ ์žฌ๊ตฌ์„ฑ ๋กœ๋ด‡์€ ๊ธฐ์กด ํ˜‘๋™๋กœ๋ด‡๋ณด๋‹ค ๋„“์€ ์•ˆ์ „ํ•œ ์ž‘์—…๊ณต๊ฐ„๊ณผ ์ž‘์—…์†๋„๋ฅผ ๊ฐ€์ง„๋‹ค.1 Introduction 1 1.1 Modularity and Recon gurability 1 1.2 Safe Interaction 4 1.3 Contributions of This Thesis 9 1.3.1 A Recon gurable Modular Robot System with Bidirectional Modules 9 1.3.2 A Modular Robot Software Programming Architecture 10 1.3.3 Anticipatory Collision Avoidance Planning 11 1.4 Organization of This Thesis 14 2 Design and Prototyping of the ModMan 17 2.1 Genderless Connector 18 2.2 Modules for ModMan 21 2.2.1 Joint Modules 21 2.2.2 Link and Gripper Modules 25 2.3 Experiments 26 2.3.1 System Setup 26 2.3.2 Repeatability Comparison with Non-recon gurable Robot Manipulators 28 2.3.3 E ect of the O set in Two-dof Modules 30 2.4 Conclusion 32 3 A Programming Architecture for Modular Recon gurable Robots 33 3.1 Data Structure for Multi-dof Joint Modules 34 3.2 Automatic Kinematic Modeling 37 3.3 Automatic Dynamic Modeling 40 3.4 Flexibility in Manipulator 42 3.5 Experiments 45 3.5.1 System Setup 46 3.5.2 Recon gurability 46 3.5.3 Pick-and-Place with Vision Sensors 48 3.6 Conclusion 49 4 A Preparatory Safety Measure for Robust Collision Avoidance 51 4.1 Preliminaries on Manipulability and Safety 52 4.2 Analysis on Reected Mass 56 4.3 Manipulability Control on S+(1;m) 60 4.3.1 Geometry of the Group of Positive Semi-de nite Matrices 60 4.3.2 Rank-One Manipulability Control 63 4.4 Collision Avoidance with Preparatory Action 65 4.4.1 Repulsive and Preparatory Potential Functions 65 4.4.2 Hierarchical Control and Task Relaxation 67 4.5 Experiments 70 4.5.1 Manipulability Control 71 4.5.2 Collision Avoidance 75 4.6 Conclusion 82 5 Collision Avoidance with Velocity-Dependent Constraints 85 5.1 Input-Output Linearization 87 5.2 Invariance Control 89 5.3 Velocity-Dependent Constraints for Robot Safety 90 5.3.1 Velocity-Dependent Repulsive Constraints 90 5.3.2 Preparatory Constraints 92 5.3.3 Corrective Control for Dangerous Initial State 93 5.4 Experiment 95 5.5 Conclusion 98 6 Conclusion 101 6.1 Overview of This Thesis 101 6.2 Future Work 104 Appendix A Appendix 107 A.1 Preliminaries on Graph Theory 107 A.2 Lie-Theoretic Formulations of Robot Kinematics and Dynamics 108 A.3 Derivatives of Eigenvectors and Eigenvalues 110 A.4 Proof of Proposition Proposition 4.1 111 A.5 Proof of Triangle Inequality When p = 1 114 A.6 Detailed Conditions for a Danger Field 115 Bibliography 117 Abstract 127Docto

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