Tutorial 3 Jacobian, Singularity, Robot Design, PDF

Tutorial 3 Jacobian, Singularity, Robot Design and Safety 1. Plan for a robot design activities stating the main robotics design phases and studies involved. Explain each of the planned activities. (20 marks). Highlight where singularity study is done in the activities. 2. Define singu...

Tutorial 3 Jacobian, Singularity, Robot Design and Safety 1. Plan for a robot design activities stating the main robotics design phases and studies involved. Explain each of the planned activities. (20 marks). Highlight where singularity study is done in the activities. 2. Define singularity in the context of robotics. Provide examples of singularities in 2- axis and 6-axis robots. (6 marks) 3. Explain the purpose of singularity protection as outlined in ISO 10218-1:2011. Why is it critical for operator safety? (6 marks) 4. Using an example of a Cartesian robot, describe the differences between joint space and Cartesian space control. How can these affect singularity behavior? (6 marks) 5. As part of design planning, using the example of a 2-axis planar robot, sketch and explain where a straight line motions from point A to point B defined in Cartesian space pass near singularities can produce high axis speeds. (6 marks) 6. Suggest design approaches for singularity protection based on compliance to Section 5.11 (a), (b) and (c) in ISO 10218-1_2011 (6 marks). Explain the consequence of the design decision (3 marks). 7. Discuss two (2) hazardous scenarios related to singularity of a robot and suggest mitigation strategies to overcome the hazard. (10 marks) 8. Derive the Jacobian matrix of a 2-axis planar robot and explain how it relates to singularity analysis where the robot might failed due to singularity. (20 marks) 9. Outline key safety considerations you would integrate into your design considering: a. Physical safety of human operators. b. Fail-safe mechanisms. c. Compliance with relevant safety standards. (6 marks) How does the safety considerations above minimize risks to human operators during operation? (6 marks) Tutorial 3 Answer 1. Robot design activities: 1. Understanding requirements – study customer and user requirements and their related engineering characteristics (2 marks) 2. Configuration design (work envelop, work volume, workspace) – plan for suitable robot configurations such as SCARA, planar or articulated (2 marks)  3. Forward Kinematics study – to study the relation between motion in Joint space in relation to position in Cartesian space without considering the force that cause the motion (2 marks) 4. Inverse Kinematics study – to study the relation between position in Cartesian space in relation to motion in Joint space without considering the force that cause the motion (2 marks) 5. Jacobian and singularity study – to study where the robot might fail at singularity event (2 marks) 6. Dynamic study – to study the robot motion when considering the forces that act on the motion (2 marks) 7. Path and trajectory planning and trajectory generation – to plan for the optimal path in cartesian space (path planning) and the motion when considering the time factor (trajectory planning). To consider how trajectory are represented in the computer after they have been planned. (2 marks) 8. Manipulator-mechanism design – including the elements of robot system i.s. manipulator (including link, actuator and sensors), end effector, external sensors and effectors, controller (2 marks) 9. Joint and Motion Control – realization of the planning (2 marks) 10. Compliance to related standards - Adherence to safety standards ensures systematic risk minimization and compliance. (2 marks) 11. Prototyping and testing – develop and verify against requirement and specification (2 marks) 2. A singularity in robotic kinematics occurs when the robot loses the ability to control its motion in certain directions, typically due to a configuration where the determinant of the Jacobian matrix becomes zero. (2 marks) a. Example in a 2-axis planar robot: When both links of the robot are fully extended or aligned, the robot cannot generate motion perpendicular to the current configuration. (2 marks)  b. Example in a 6-axis robot: When the wrist axes align, leading to a "wrist singularity," the robot loses one degree of freedom, causing abrupt motion changes. (2 marks) 3. Singularity protection a. prevents hazardous robot motions caused by high-speed axis movements near singularities. (2 marks) b. This ensures the safety of operators by either stopping the motion, limiting speed, or ensuring controlled passage through the singularity without creating dangerous conditions. (2 marks) c. It is critical because unexpected high-speed movements near singularities can lead to operator injuries or equipment damage. (2 marks) 4. Joint and space control has the following characteristics in addressing singularities issues: a. Joint space control: Commands the individual joint angles or positions directly. Motion is defined by how each joint moves. Singularities may not be encountered directly but can cause erratic path motion. (2 marks) b. Cartesian space control: Commands the end effector to move along a defined path in Cartesian coordinates. Near singularities, small changes in end-effector position require large, rapid joint movements. (2 marks) c. For example, in a Cartesian robot moving along a straight line near a singularity, high joint velocities may be required to maintain the Cartesian trajectory, creating instability and safety risks. (2 marks) 5. Using example of a 2 dof planar robot: a. A manipulator is following a Cartesian straight-line path and approaches a singular configuration of the mechanism, one or more joint velocities might increase toward infinity. (2 marks) b. All points along the path are reachable, but as the robot goes past the middle portion of the path ((When 𝜃2 near 0 or 180 degrees), the velocity of joint one is very high. (2 marks) c. Figure below show a sketch illustrating the situation. (2 marks) 6. Singularity safety design choice and consequence: Section 5.11 in ISO 10218-1_2011 (Robots and Robotic Devices – Safety requirements for industrial robots) states the following: “5.11 Singularity protection Motions defined in Cartesian space that pass near singularities can produce high axis speeds. These high speeds can be unexpected to an operator. When in the manual reduced-speed mode or hand guiding, the robot control shall do one of the following: a) stop robot motion and provide a warning prior to the robot passing through or correcting for a singularity during coordinated motion (control wherein the axes of the robot arrive at their respective end points simultaneously, giving a smooth appearance to the motion and control wherein the motions of the axes are such that the TCP moves along a prescribed path) initiated from the teach pendant, or b) generate an audible or visible warning signal and continue to pass through the singularity with the velocity of each link of the robot arm limited to a maximum speed of 250 mm/s, or c) in the case that the singularity can be controlled without creating any hazardous motion, no additional protection is required.” (Note: Hazard is a condition, situation, or source with the potential to cause harm.) a. According to option (a) in 5.11, i. stop robot motion and provide a warning prior to the robot passing through or correcting for a singularity during coordinated motion. (1 mark)  ii. The robot programming can have a condition to stop when robot joint 2 reach certain angle (1 mark) iii. The consequence, the continuous motion cannot be observed (1 mark) b. According to option (b) in 5.11, i. generate an audible or visible warning signal and continue to pass through the singularity with the velocity of each link of the robot arm limited to a maximum speed of 250 mm/s (1 mark) ii. The robot programming can have a condition to produce alarm when robot joint 2 reach certain angle (1 mark) iii. The consequence, the desired temporal attributes of the path might be lost, but at least the spatial aspect of the trajectory definition is adhered to. (1 mark) c. According to option (c) in 5.11 i. in the case that the singularity can be controlled without creating any hazardous motion, no additional protection is required. (Note: Hazard is a condition, situation, or source with the potential to cause harm.) (1 mark) ii. One approach where this can happen, is scaling down the overall velocity of the path to a speed where all joints stay within their velocity capabilities. (1 mark) iii. The consequence, the desired temporal attributes of the path might be lost, but at least the spatial aspect of the trajectory definition is adhered to. (1 mark) 7. Hazard is a condition, situation, or source with the potential to cause harm. Here are several hazardous scenarios related to singularity of a robot. (Each Hazard – 3 marks, each mitigation – 2 marks) Mechanical Hazards Unpredictable Movements: Near singularities, joint velocities can become extremely high, leading to erratic and uncontrollable robot movements that could harm nearby operators or equipment. Example: A robotic arm moving too quickly and colliding with its environment.  Excessive Strain on Joints: Singularities can cause uneven distribution of forces, placing excessive strain on certain joints, leading to mechanical failure or breakage. Example: Damage to servo motors or actuators. Control Hazards Loss of Position Control: In a singularity, the robot might lose precise control over its end-effector position, resulting in inaccurate movements or unintended collisions. Example: A pick-and-place robot missing its target and dropping objects. Inverted Kinematics Failure: Calculations for joint angles based on Cartesian path requirements may fail, causing unexpected robot behavior. Example: A robot arm freezing or deviating off course in a manufacturing line. Software Hazards Instability in Path Planning Algorithms: Singularities can lead to errors in path interpolation, causing abrupt or unsafe movements. Example: A robot halting mid-task or executing unsafe trajectories. Sensor Misinterpretation: Encoders and sensors might misread positions due to rapid joint movement near singularities, leading to feedback errors. Example: Incorrect signals triggering unintended responses. Ergonomic Hazards Operator Interaction: Singularities can cause unpredictable robot movements, increasing the risk of injury for operators working nearby. Example: A robot arm unexpectedly swinging close to a technician. Electrical Hazards Overloading Electrical Systems: High joint velocities during singularities can draw excessive current, potentially overloading circuits or damaging components. Example: Actuator overheating or control system failure. Safety System Hazards  Failure of Emergency Protocols: Safety mechanisms like collision avoidance systems may not respond effectively if singularity-induced movements occur too quickly. Example: Safety stops failing to engage in time. Mitigation Strategies Path Planning: Avoid configurations that bring the robot near singularities by using advanced algorithms and trajectory optimization. Joint Limit Enforcement: Limit joint velocities and accelerations to prevent unsafe behaviors near singularities. Redundancy: Employ redundant degrees of freedom (e.g., 7-DOF arms) to navigate around singular configurations. Simulation and Testing: Use simulation tools to identify and avoid singularities in the design phase. Operator Training: Educate operators on recognizing and managing risks associated with singularities. 8. The Jacobian matrix for a 2-axis planar robot is given by: angular velocity of link 𝑖 + 1 with respect to frame {𝑖 + 1}: 𝑖+1 𝜔𝑖+1 = 𝑖+1 𝑖 𝑖 𝑅 𝜔𝑖 + 𝜃̇𝑖+1 𝑖+1𝑍̂𝑖+1 linear velocity of the origin of frame {𝑖 + 1}: 𝑖+1 𝑖 𝑣𝑖+1 = 𝑖+1 𝑖 𝑅 ( 𝑣𝑖 + 𝑖𝜔𝑖 × 𝑖𝑃𝑖+1 ) link transformations of the robot in are 𝑐𝜃1 −𝑠𝜃1 0 0 𝑐𝜃2 −𝑠𝜃2 0 𝑙1 1 0 0 𝑙2 0 𝑠𝜃1 𝑐𝜃1 0 0 1 𝑠𝜃 𝑐𝜃2 0 0 2 0 1 0 0 1𝑇 = [ ] , 2𝑇 = [ 2 ] , 3𝑇 = [ ] 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 Using link to link velocity propagation method: 𝑓𝑜𝑟 𝑖 = 0,  0 1 𝜔1 = [ 0 ] 𝜃̇1 0 1 𝑣1 = 0 𝑓𝑜𝑟 𝑖 = 1, 2 𝜔2 = 21𝑅 1𝜔1 + 𝜃̇2 2𝑍̂2 𝐶2 𝑆2 0 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 21𝑅 = 12𝑅−1 = [−𝑆2 𝐶2 0] 0 0 1 𝐶2 𝑆2 0 0 0 2 1 1𝑅 𝜔1 = [−𝑆2 𝐶2 0] [ ] = [ 0 ] 0 0 0 1 𝜃̇1 𝜃̇1 0 0 0 2 𝜔2 = [ 0 ] + [ 0 ] = [ 0 ] 𝜃̇1 𝜃̇2 𝜃̇1 + 𝜃̇2 𝑁𝑒𝑥𝑡, 2𝑣2 = 21𝑅( 1𝑣1 + 1𝜔1 × 1𝑃2 ) 0 𝑙1 0 1 𝜔1 × 𝑃2 = [ ] × [ 0 ] = [𝑙1 𝜃̇1 ] 1 0 𝜃̇1 0 0 𝑖 −𝑗 𝑘 0 𝑤ℎ𝑒𝑟𝑒 [ 0 0 𝜃̇1 ] = 0𝑖 − 𝑙1 𝜃̇1 𝑗 + 0𝑘 = [𝑙1 𝜃̇1 ] 𝑙1 0 0 0 𝐶2 𝑆2 0 0 𝑙1 𝑆2 𝜃̇1 2 𝑣2 = [−𝑆2 𝐶2 0] [𝑙1 𝜃̇1 ] = [𝑙1 𝑐2 𝜃̇1 ] 0 0 1 0 0 𝑓𝑜𝑟 𝑖 = 2, 𝑖+1 𝜔𝑖+1 = 𝑖+1 𝑖 𝑖 𝑅 𝜔𝑖 + 𝜃̇𝑖+1 𝑖+1𝑍̂𝑖+1 3 𝜔3 = 32𝑅 2𝜔2 + 𝜃̇3 3𝑍̂3 1 0 0 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 32𝑅 = 23𝑅 −1 = [0 1 0] 0 0 1  1 0 0 0 0 3 2 2 𝑅 𝜔2 = [0 1 0] [ 0 ] = [ 0 ] 0 0 1 𝜃̇1 + 𝜃̇2 𝜃̇1 + 𝜃̇2 𝜃̇3 3𝑍̂3 = 0 3 𝜔3 = 2 𝜔2 𝑁𝑒𝑥𝑡, 3𝑣3 = 32𝑅( 2𝑣2 + 2𝜔2 × 2𝑃3 ) 0 𝑙2 0 2 𝜔2 × 𝑃3 = [ 0 ] × [ 0 ] = [𝑙2 (𝜃̇1 + 𝜃̇2 )] 2 𝜃̇1 + 𝜃̇2 0 0 𝑖 −𝑗 𝑘 0 𝑤ℎ𝑒𝑟𝑒 [ 0 0 ̇𝜃1 + 𝜃̇2 ] = 0𝑖 − 𝑙2 (𝜃̇1 + 𝜃̇2 )𝑗 + 0𝑘 = [𝑙2 (𝜃̇1 + 𝜃̇2 )] 𝑙2 0 0 0 1 0 0 𝑙1 𝑆2 𝜃̇1 0 3 𝑣3 = [0 1 0] ([𝑙1 𝑐2 𝜃̇1 ] + [𝑙2 (𝜃1 + 𝜃̇2 )]) ̇ 0 0 1 0 0 𝑙1 𝑠2 𝜃̇1 3 𝑣3 = [𝑙1 𝑐2 𝜃̇1 + 𝑙2 (𝜃̇1 + 𝜃̇2 )] 0 (𝑙1 𝑠2 )𝜃̇1 + (0)𝜃̇2 3 𝑣3 = [(𝑙1 𝑐2 + 𝑙2 )𝜃̇1 + (𝑙2 )𝜃̇2 ] 0 3 𝑣 = 3𝐽 (𝜃)𝜃̇ The Jacobian is highlighted: 𝑣1 𝑙 𝑠 0 𝜃̇1 [𝑣 ] = [ 1 2 ][ ] 2 𝑙1 2 + 𝑙2 𝑐 𝑙2 𝜃̇2 𝑣1 𝑎 𝑏 𝜃̇1 [𝑣 ] = [ ][ ] 2 𝑐 𝑑 𝜃̇2 𝜃̇ 1 𝑎 𝑏 [ 1] = [ ]? 𝜃̇2 𝑎𝑑 − 𝑏𝑐 𝑐 𝑑  (10 marks) Singularities occurs when 𝐷𝐸𝑇[𝐽(𝜃)] = 0 𝐷𝐸𝑇[𝐽(𝜃)] = 𝑎𝑑 − 𝑏𝑐 𝐷𝐸𝑇[𝐽(𝜃)] = 𝑙1 𝑙2 𝑠2 𝑙1 𝑙2 𝑠2 = 0 sin (𝜃2 ) = 0 𝜃2 = sin−1(0) = 0° 𝑜𝑟 180° If 𝜃2 = 0 𝑣 𝑣 Then 𝜃̇1 = 𝑙 𝑠1 = 01 = ∞ 1 2 Joint rate approach infinity as 𝜃2 → 0 𝑜𝑟 180° (10 marks) 9. Safety considerations: a. Physical Safety of Human Operators (2 marks) i. Rounded edges and soft materials in the robot's design to minimize injury upon contact. ii. Limited payload and speed to reduce the impact force. b. Fail-Safe Mechanisms (2 marks) i. Emergency stop button and software kill switches. ii. Built-in sensors for collision detection to immediately stop movement upon contact. iii. Redundant systems to prevent failure of critical safety functions. c. Compliance with Standards (2 marks) i. Adherence to ISO 10218-1 and ISO/TS 15066 standards for collaborative robots. ii. Proper labeling and documentation for safety procedures.  iii. Compliance with local workplace safety regulations. The safety considerations minimize risks to human operators during operation by: a. Physical Safety of Human Operators: Rounded edges and soft materials reduce injury risks during unintended collisions. (2 marks) b. Fail-Safe Mechanisms: Emergency stop buttons allow operators to halt the robot immediately in dangerous situations. Collision detection sensors enhance real-time responsiveness to avoid accidents. (2 marks) c. Compliance with Standards: Adherence to safety standards ensures systematic risk minimization and compliance. (2 marks) End