# coding=utf-8 from abc import ABC, abstractmethod from sympy.core.numbers import pi from sympy.physics.mechanics.body import Body from sympy.physics.vector import Vector, dynamicsymbols, cross from sympy.physics.vector.frame import ReferenceFrame import warnings __all__ = ['Joint', 'PinJoint', 'PrismaticJoint'] class Joint(ABC): """Abstract base class for all specific joints. Explanation =========== A joint subtracts degrees of freedom from a body. This is the base class for all specific joints and holds all common methods acting as an interface for all joints. Custom joint can be created by inheriting Joint class and defining all abstract functions. The abstract methods are: - ``_generate_coordinates`` - ``_generate_speeds`` - ``_orient_frames`` - ``_set_angular_velocity`` - ``_set_linar_velocity`` Parameters ========== name : string A unique name for the joint. parent : Body The parent body of joint. child : Body The child body of joint. coordinates: List of dynamicsymbols, optional Generalized coordinates of the joint. speeds : List of dynamicsymbols, optional Generalized speeds of joint. parent_joint_pos : Vector, optional Vector from the parent body's mass center to the point where the parent and child are connected. The default value is the zero vector. child_joint_pos : Vector, optional Vector from the child body's mass center to the point where the parent and child are connected. The default value is the zero vector. parent_axis : Vector, optional Axis fixed in the parent body which aligns with an axis fixed in the child body. The default is x axis in parent's reference frame. child_axis : Vector, optional Axis fixed in the child body which aligns with an axis fixed in the parent body. The default is x axis in child's reference frame. Attributes ========== name : string The joint's name. parent : Body The joint's parent body. child : Body The joint's child body. coordinates : list List of the joint's generalized coordinates. speeds : list List of the joint's generalized speeds. parent_point : Point The point fixed in the parent body that represents the joint. child_point : Point The point fixed in the child body that represents the joint. parent_axis : Vector The axis fixed in the parent frame that represents the joint. child_axis : Vector The axis fixed in the child frame that represents the joint. kdes : list Kinematical differential equations of the joint. Notes ===== The direction cosine matrix between the child and parent is formed using a simple rotation about an axis that is normal to both ``child_axis`` and ``parent_axis``. In general, the normal axis is formed by crossing the ``child_axis`` into the ``parent_axis`` except if the child and parent axes are in exactly opposite directions. In that case the rotation vector is chosen using the rules in the following table where ``C`` is the child reference frame and ``P`` is the parent reference frame: .. list-table:: :header-rows: 1 * - ``child_axis`` - ``parent_axis`` - ``rotation_axis`` * - ``-C.x`` - ``P.x`` - ``P.z`` * - ``-C.y`` - ``P.y`` - ``P.x`` * - ``-C.z`` - ``P.z`` - ``P.y`` * - ``-C.x-C.y`` - ``P.x+P.y`` - ``P.z`` * - ``-C.y-C.z`` - ``P.y+P.z`` - ``P.x`` * - ``-C.x-C.z`` - ``P.x+P.z`` - ``P.y`` * - ``-C.x-C.y-C.z`` - ``P.x+P.y+P.z`` - ``(P.x+P.y+P.z) × P.x`` """ def __init__(self, name, parent, child, coordinates=None, speeds=None, parent_joint_pos=None, child_joint_pos=None, parent_axis=None, child_axis=None): if not isinstance(name, str): raise TypeError('Supply a valid name.') self._name = name if not isinstance(parent, Body): raise TypeError('Parent must be an instance of Body.') self._parent = parent if not isinstance(child, Body): raise TypeError('Parent must be an instance of Body.') self._child = child self._coordinates = self._generate_coordinates(coordinates) self._speeds = self._generate_speeds(speeds) self._kdes = self._generate_kdes() self._parent_axis = self._axis(parent, parent_axis) self._child_axis = self._axis(child, child_axis) self._parent_point = self._locate_joint_pos(parent, parent_joint_pos) self._child_point = self._locate_joint_pos(child, child_joint_pos) self._orient_frames() self._set_angular_velocity() self._set_linear_velocity() def __str__(self): return self.name def __repr__(self): return self.__str__() @property def name(self): return self._name @property def parent(self): """Parent body of Joint.""" return self._parent @property def child(self): """Child body of Joint.""" return self._child @property def coordinates(self): """List generalized coordinates of the joint.""" return self._coordinates @property def speeds(self): """List generalized coordinates of the joint..""" return self._speeds @property def kdes(self): """Kinematical differential equations of the joint.""" return self._kdes @property def parent_axis(self): """The axis of parent frame.""" return self._parent_axis @property def child_axis(self): """The axis of child frame.""" return self._child_axis @property def parent_point(self): """The joint's point where parent body is connected to the joint.""" return self._parent_point @property def child_point(self): """The joint's point where child body is connected to the joint.""" return self._child_point @abstractmethod def _generate_coordinates(self, coordinates): """Generate list generalized coordinates of the joint.""" pass @abstractmethod def _generate_speeds(self, speeds): """Generate list generalized speeds of the joint.""" pass @abstractmethod def _orient_frames(self): """Orient frames as per the joint.""" pass @abstractmethod def _set_angular_velocity(self): pass @abstractmethod def _set_linear_velocity(self): pass def _generate_kdes(self): kdes = [] t = dynamicsymbols._t for i in range(len(self.coordinates)): kdes.append(-self.coordinates[i].diff(t) + self.speeds[i]) return kdes def _axis(self, body, ax): if ax is None: ax = body.frame.x return ax if not isinstance(ax, Vector): raise TypeError("Axis must be of type Vector.") if not ax.dt(body.frame) == 0: msg = ('Axis cannot be time-varying when viewed from the ' 'associated body.') raise ValueError(msg) return ax def _locate_joint_pos(self, body, joint_pos): if joint_pos is None: joint_pos = Vector(0) if not isinstance(joint_pos, Vector): raise ValueError('Joint Position must be supplied as Vector.') if not joint_pos.dt(body.frame) == 0: msg = ('Position Vector cannot be time-varying when viewed from ' 'the associated body.') raise ValueError(msg) point_name = self._name + '_' + body.name + '_joint' return body.masscenter.locatenew(point_name, joint_pos) def _alignment_rotation(self, parent, child): # Returns the axis and angle between two axis(vectors). angle = parent.angle_between(child) axis = cross(child, parent).normalize() return angle, axis def _generate_vector(self): parent_frame = self.parent.frame components = self.parent_axis.to_matrix(parent_frame) x, y, z = components[0], components[1], components[2] if x != 0: if y!=0: if z!=0: return cross(self.parent_axis, parent_frame.x) if z!=0: return parent_frame.y return parent_frame.z if x == 0: if y!=0: if z!=0: return parent_frame.x return parent_frame.x return parent_frame.y def _set_orientation(self): #Helper method for `orient_axis()` self.child.frame.orient_axis(self.parent.frame, self.parent_axis, 0) angle, axis = self._alignment_rotation(self.parent_axis, self.child_axis) with warnings.catch_warnings(): warnings.filterwarnings("ignore", category=UserWarning) if axis != Vector(0) or angle == pi: if angle == pi: axis = self._generate_vector() int_frame = ReferenceFrame('int_frame') int_frame.orient_axis(self.child.frame, self.child_axis, 0) int_frame.orient_axis(self.parent.frame, axis, angle) return int_frame return self.parent.frame class PinJoint(Joint): """Pin (Revolute) Joint. .. image:: PinJoint.png Explanation =========== A pin joint is defined such that the joint rotation axis is fixed in both the child and parent and the location of the joint is relative to the mass center of each body. The child rotates an angle, θ, from the parent about the rotation axis and has a simple angular speed, ω, relative to the parent. The direction cosine matrix between the child and parent is formed using a simple rotation about an axis that is normal to both ``child_axis`` and ``parent_axis``, see the Notes section for a detailed explanation of this. Parameters ========== name : string A unique name for the joint. parent : Body The parent body of joint. child : Body The child body of joint. coordinates: dynamicsymbol, optional Generalized coordinates of the joint. speeds : dynamicsymbol, optional Generalized speeds of joint. parent_joint_pos : Vector, optional Vector from the parent body's mass center to the point where the parent and child are connected. The default value is the zero vector. child_joint_pos : Vector, optional Vector from the child body's mass center to the point where the parent and child are connected. The default value is the zero vector. parent_axis : Vector, optional Axis fixed in the parent body which aligns with an axis fixed in the child body. The default is x axis in parent's reference frame. child_axis : Vector, optional Axis fixed in the child body which aligns with an axis fixed in the parent body. The default is x axis in child's reference frame. Attributes ========== name : string The joint's name. parent : Body The joint's parent body. child : Body The joint's child body. coordinates : list List of the joint's generalized coordinates. speeds : list List of the joint's generalized speeds. parent_point : Point The point fixed in the parent body that represents the joint. child_point : Point The point fixed in the child body that represents the joint. parent_axis : Vector The axis fixed in the parent frame that represents the joint. child_axis : Vector The axis fixed in the child frame that represents the joint. kdes : list Kinematical differential equations of the joint. Examples ========= A single pin joint is created from two bodies and has the following basic attributes: >>> from sympy.physics.mechanics import Body, PinJoint >>> parent = Body('P') >>> parent P >>> child = Body('C') >>> child C >>> joint = PinJoint('PC', parent, child) >>> joint PinJoint: PC parent: P child: C >>> joint.name 'PC' >>> joint.parent P >>> joint.child C >>> joint.parent_point PC_P_joint >>> joint.child_point PC_C_joint >>> joint.parent_axis P_frame.x >>> joint.child_axis C_frame.x >>> joint.coordinates [theta_PC(t)] >>> joint.speeds [omega_PC(t)] >>> joint.child.frame.ang_vel_in(joint.parent.frame) omega_PC(t)*P_frame.x >>> joint.child.frame.dcm(joint.parent.frame) Matrix([ [1, 0, 0], [0, cos(theta_PC(t)), sin(theta_PC(t))], [0, -sin(theta_PC(t)), cos(theta_PC(t))]]) >>> joint.child_point.pos_from(joint.parent_point) 0 To further demonstrate the use of the pin joint, the kinematics of simple double pendulum that rotates about the Z axis of each connected body can be created as follows. >>> from sympy import symbols, trigsimp >>> from sympy.physics.mechanics import Body, PinJoint >>> l1, l2 = symbols('l1 l2') First create bodies to represent the fixed ceiling and one to represent each pendulum bob. >>> ceiling = Body('C') >>> upper_bob = Body('U') >>> lower_bob = Body('L') The first joint will connect the upper bob to the ceiling by a distance of ``l1`` and the joint axis will be about the Z axis for each body. >>> ceiling_joint = PinJoint('P1', ceiling, upper_bob, ... child_joint_pos=-l1*upper_bob.frame.x, ... parent_axis=ceiling.frame.z, ... child_axis=upper_bob.frame.z) The second joint will connect the lower bob to the upper bob by a distance of ``l2`` and the joint axis will also be about the Z axis for each body. >>> pendulum_joint = PinJoint('P2', upper_bob, lower_bob, ... child_joint_pos=-l2*lower_bob.frame.x, ... parent_axis=upper_bob.frame.z, ... child_axis=lower_bob.frame.z) Once the joints are established the kinematics of the connected bodies can be accessed. First the direction cosine matrices of pendulum link relative to the ceiling are found: >>> upper_bob.frame.dcm(ceiling.frame) Matrix([ [ cos(theta_P1(t)), sin(theta_P1(t)), 0], [-sin(theta_P1(t)), cos(theta_P1(t)), 0], [ 0, 0, 1]]) >>> trigsimp(lower_bob.frame.dcm(ceiling.frame)) Matrix([ [ cos(theta_P1(t) + theta_P2(t)), sin(theta_P1(t) + theta_P2(t)), 0], [-sin(theta_P1(t) + theta_P2(t)), cos(theta_P1(t) + theta_P2(t)), 0], [ 0, 0, 1]]) The position of the lower bob's masscenter is found with: >>> lower_bob.masscenter.pos_from(ceiling.masscenter) l1*U_frame.x + l2*L_frame.x The angular velocities of the two pendulum links can be computed with respect to the ceiling. >>> upper_bob.frame.ang_vel_in(ceiling.frame) omega_P1(t)*C_frame.z >>> lower_bob.frame.ang_vel_in(ceiling.frame) omega_P1(t)*C_frame.z + omega_P2(t)*U_frame.z And finally, the linear velocities of the two pendulum bobs can be computed with respect to the ceiling. >>> upper_bob.masscenter.vel(ceiling.frame) l1*omega_P1(t)*U_frame.y >>> lower_bob.masscenter.vel(ceiling.frame) l1*omega_P1(t)*U_frame.y + l2*(omega_P1(t) + omega_P2(t))*L_frame.y """ def __init__(self, name, parent, child, coordinates=None, speeds=None, parent_joint_pos=None, child_joint_pos=None, parent_axis=None, child_axis=None): super().__init__(name, parent, child, coordinates, speeds, parent_joint_pos, child_joint_pos, parent_axis, child_axis) def __str__(self): return (f'PinJoint: {self.name} parent: {self.parent} ' f'child: {self.child}') def _generate_coordinates(self, coordinate): coordinates = [] if coordinate is None: theta = dynamicsymbols('theta' + '_' + self._name) coordinate = theta coordinates.append(coordinate) return coordinates def _generate_speeds(self, speed): speeds = [] if speed is None: omega = dynamicsymbols('omega' + '_' + self._name) speed = omega speeds.append(speed) return speeds def _orient_frames(self): frame = self._set_orientation() self.child.frame.orient_axis(frame, self.parent_axis, self.coordinates[0]) def _set_angular_velocity(self): self.child.frame.set_ang_vel(self.parent.frame, self.speeds[0] * self.parent_axis.normalize()) def _set_linear_velocity(self): self.parent_point.set_vel(self.parent.frame, 0) self.child_point.set_vel(self.child.frame, 0) self.child_point.set_pos(self.parent_point, 0) self.child.masscenter.v2pt_theory(self.parent_point, self.parent.frame, self.child.frame) class PrismaticJoint(Joint): """Prismatic (Sliding) Joint. .. image:: PrismaticJoint.png Explanation =========== It is defined such that the child body translates with respect to the parent body along the body fixed parent axis. The location of the joint is defined by two points in each body which coincides when the generalized coordinate is zero. The direction cosine matrix between the child and parent is formed using a simple rotation about an axis that is normal to both ``child_axis`` and ``parent_axis``, see the Notes section for a detailed explanation of this. Parameters ========== name : string A unique name for the joint. parent : Body The parent body of joint. child : Body The child body of joint. coordinates: dynamicsymbol, optional Generalized coordinates of the joint. speeds : dynamicsymbol, optional Generalized speeds of joint. parent_joint_pos : Vector, optional Vector from the parent body's mass center to the point where the parent and child are connected. The default value is the zero vector. child_joint_pos : Vector, optional Vector from the child body's mass center to the point where the parent and child are connected. The default value is the zero vector. parent_axis : Vector, optional Axis fixed in the parent body which aligns with an axis fixed in the child body. The default is x axis in parent's reference frame. child_axis : Vector, optional Axis fixed in the child body which aligns with an axis fixed in the parent body. The default is x axis in child's reference frame. Attributes ========== name : string The joint's name. parent : Body The joint's parent body. child : Body The joint's child body. coordinates : list List of the joint's generalized coordinates. speeds : list List of the joint's generalized speeds. parent_point : Point The point fixed in the parent body that represents the joint. child_point : Point The point fixed in the child body that represents the joint. parent_axis : Vector The axis fixed in the parent frame that represents the joint. child_axis : Vector The axis fixed in the child frame that represents the joint. kdes : list Kinematical differential equations of the joint. Examples ========= A single prismatic joint is created from two bodies and has the following basic attributes: >>> from sympy.physics.mechanics import Body, PrismaticJoint >>> parent = Body('P') >>> parent P >>> child = Body('C') >>> child C >>> joint = PrismaticJoint('PC', parent, child) >>> joint PrismaticJoint: PC parent: P child: C >>> joint.name 'PC' >>> joint.parent P >>> joint.child C >>> joint.parent_point PC_P_joint >>> joint.child_point PC_C_joint >>> joint.parent_axis P_frame.x >>> joint.child_axis C_frame.x >>> joint.coordinates [x_PC(t)] >>> joint.speeds [v_PC(t)] >>> joint.child.frame.ang_vel_in(joint.parent.frame) 0 >>> joint.child.frame.dcm(joint.parent.frame) Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> joint.child_point.pos_from(joint.parent_point) x_PC(t)*P_frame.x To further demonstrate the use of the prismatic joint, the kinematics of two masses sliding, one moving relative to a fixed body and the other relative to the moving body. about the X axis of each connected body can be created as follows. >>> from sympy.physics.mechanics import PrismaticJoint, Body First create bodies to represent the fixed ceiling and one to represent a particle. >>> wall = Body('W') >>> Part1 = Body('P1') >>> Part2 = Body('P2') The first joint will connect the particle to the ceiling and the joint axis will be about the X axis for each body. >>> J1 = PrismaticJoint('J1', wall, Part1) The second joint will connect the second particle to the first particle and the joint axis will also be about the X axis for each body. >>> J2 = PrismaticJoint('J2', Part1, Part2) Once the joint is established the kinematics of the connected bodies can be accessed. First the direction cosine matrices of Part relative to the ceiling are found: >>> Part1.dcm(wall) Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> Part2.dcm(wall) Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) The position of the particles' masscenter is found with: >>> Part1.masscenter.pos_from(wall.masscenter) x_J1(t)*W_frame.x >>> Part2.masscenter.pos_from(wall.masscenter) x_J1(t)*W_frame.x + x_J2(t)*P1_frame.x The angular velocities of the two particle links can be computed with respect to the ceiling. >>> Part1.ang_vel_in(wall) 0 >>> Part2.ang_vel_in(wall) 0 And finally, the linear velocities of the two particles can be computed with respect to the ceiling. >>> Part1.masscenter_vel(wall) v_J1(t)*W_frame.x >>> Part2.masscenter.vel(wall.frame) v_J1(t)*W_frame.x + Derivative(x_J2(t), t)*P1_frame.x """ def __init__(self, name, parent, child, coordinates=None, speeds=None, parent_joint_pos=None, child_joint_pos=None, parent_axis=None, child_axis=None): super().__init__(name, parent, child, coordinates, speeds, parent_joint_pos, child_joint_pos, parent_axis, child_axis) def __str__(self): return (f'PrismaticJoint: {self.name} parent: {self.parent} ' f'child: {self.child}') def _generate_coordinates(self, coordinate): coordinates = [] if coordinate is None: x = dynamicsymbols('x' + '_' + self._name) coordinate = x coordinates.append(coordinate) return coordinates def _generate_speeds(self, speed): speeds = [] if speed is None: y = dynamicsymbols('v' + '_' + self._name) speed = y speeds.append(speed) return speeds def _orient_frames(self): frame = self._set_orientation() self.child.frame.orient_axis(frame, self.parent_axis, 0) def _set_angular_velocity(self): self.child.frame.set_ang_vel(self.parent.frame, 0) def _set_linear_velocity(self): self.parent_point.set_vel(self.parent.frame, 0) self.child_point.set_vel(self.child.frame, 0) self.child_point.set_pos(self.parent_point, self.coordinates[0] * self.parent_axis.normalize()) self.child_point.set_vel(self.parent.frame, self.speeds[0] * self.parent_axis.normalize()) self.child.masscenter.set_vel(self.parent.frame, self.speeds[0] * self.parent_axis.normalize())