What is Robotic Kinematics

Robotic Kinematics and Dynamics

Content of Kinematics and Dynamics

• What is Robotics

• Robot Kinematics and Dynamics

• Robot Control (PID, Model Predictive Control)

• Perception in Robotics (Sensors, SLAM)

• Robot Planning and Decision-Making

• Expert Systems

1. What is Robotics?

Robotics is a branch of engineering and computer science that deals with the design, construction, operation, and application of robots. It involves the study of various aspects such as mechanics, electronics, and programming to create machines that can perform various tasks autonomously or with human assistance.

2. Robot Kinematics and Dynamics:

Robot kinematics deals with the study of the motion and position of robots without considering the forces that cause the motion. On the other hand, robot dynamics deals with the study of the forces that cause motion in robots. It involves the use of equations and models to describe the behaviour of robots.

3. Robot Control:

Robot control involves the use of algorithms and techniques to control the motion of robots. Two common types of robot control are PID (Proportional-Integral-Derivative) control and Model Predictive Control (MPC).

PID Control:

PID control is a feedback control technique that is widely used in robotics. It involves calculating the error between the desired output and the actual output and adjusting the control parameters based on the error. The control parameters include the proportional, integral, and derivative gains, which determine the response of the system.

Example Python code for implementing PID control in robotics:

Python Code

import numpy as np

class PIDController:

    def __init__(self, Kp, Ki, Kd, setpoint):

        self.Kp = Kp

        self.Ki = Ki

        self.Kd = Kd

        self.setpoint = setpoint

        self.integral = 0

        self.previous_error = 0

    def control(self, measurement):

        error = self.setpoint - measurement

        self.integral += error

        derivative = error - self.previous_error

      output = self.Kp*error + self.Ki*self.integral +                                                                                  self.Kd*derivative

        self.previous_error = error

        return output

# Example usage

controller = PIDController(Kp=1, Ki=0.1, Kd=0.05, setpoint=0)

measurement = 0.5

while True:

    output = controller.control(measurement)

    measurement += output

Model Predictive Control (MPC):

MPC is a control technique that involves predicting the future behaviour of the system and optimizing a cost function based on the predicted behaviour. It is commonly used in robotics for trajectory planning and control.

4. Perception in Robotics:

Perception in robotics involves the use of sensors to gather information about the environment and the robot itself. Two common types of sensors used in robotics are LIDAR (Light Detection and Model Predictive Control (MPC).

MPC is a control technique that involves predicting the future behaviour of the system and optimizing a cost function based on the predicted behaviour. It is commonly used in robotics for trajectory planning and control.

Here's an example of how to implement MPC in Python using the cvxpy library:

python code

import cvxpy as cp

import numpy as np

# Define system dynamics

A = np.array([[1.0, 0.1], [0.0, 1.0]])

B = np.array([[0.005], [0.1]])

# Define cost function

Q = np.diag([1.0, 1.0])

R = np.diag([0.01])

P = np.diag([1.0, 1.0])

# Define MPC parameters

N = 10

x0 = np.array([0.0, 0.0])

# Define optimization variables

x = cp.Variable((2, N+1))

u = cp.Variable((1, N))

# Define constraints and objective function

constraints = []

objective = cp.quad_form(x[:,N], P)

for i in range(N):

    objective += cp.quad_form(x[:,i], Q) + cp.quad_form(u[:,i], R)

    constraints += [x[:,i+1] == A@x[:,i] + B@u[:,i], cp.norm(u[:,i], 'inf') <= 1.0]

constraints += [x[:,0] == x0]

problem = cp.Problem(cp.Minimize(objective), constraints)

solver = cp.ECOS_BB()

solution = solver.solve(problem)

# Extract control inputs

u_star = u.value[0,:]

# Apply control inputs to the system

x_traj = np.zeros((2, N+1))

x_traj[:,0] = x0

for i in range(N):

    x_traj[:,i+1] = A@x_traj[:,i] + B@u_star[i]

Here's an example of how to use OpenCV to perform object detection on a video stream:

python code

import cv2

# Load classifier for object detection

classifier = cv2.CascadeClassifier('haarcascade_frontalface_default.xml')

# Open video stream

cap = cv2.VideoCapture(0)

while True:

    # Read a frame from a video stream

    ret, frame = cap.read()

    # Convert to grayscale

    gray = cv2.cvtColor(frame, cv2.COLOR_BGR2GRAY)

    # Detect objects in the image

    objects = classifier.detectMultiScale(gray, scaleFactor=1.1, minNeighbors=5, minSize=(30, 30))

    # Draw bounding boxes around objects

    for (x, y, w, h) in objects:

        cv2.rectangle(frame, (x, y), (x+w, y+h), (0, 255, 0), 2)

    # Display result

    cv2.imshow('frame', frame)

    # Wait for user input

    if cv2.waitKey(1) & 0xFF == ord('q'):

        break

# Release video stream and close window

cap.release()

cv2.destroyAllWindows()

This code uses a pre-trained Haar Cascade classifier to detect faces in a video stream. It converts each frame to grayscale, detects objects using the classifier, and draws bounding boxes around them. The resulting video stream is displayed in a window, and the program exits when the user perception in robotics involves the use of sensors to gather information about the environment and the robot itself. Two common types of sensors used in robotics are LIDAR (Light Detection and Ranging) and RGB-D cameras (cameras that capture both colour and depth information).

SLAM 

(Simultaneous Localization and Mapping) is a popular technique used in robotics for building maps of an unknown environment while simultaneously keeping track of the robot's location within that environment? Here is an example of how to implement a basic SLAM algorithm using Python and the OpenCV library:

python code

import cv2

import numpy as np

# Initialize map

map = np.zeros((200, 200), dtype=np.uint8)

# Initialize robot position

x = 100

y = 100

# Loop over frames

for i in range(100):

    # Simulate movement

    x += 2

    y += 2

    # Capture frame

    frame = np.zeros((200, 200), dtype=np.uint8)

    cv2.circle(frame, (x, y), 10, 255, -1)

    # Update map

    map[frame > 0] = 255

    # Display map

    cv2.imshow('Map', map)

    cv2.waitKey(10)

cv2.destroyAllWindows()

In this example, we simulate a robot moving through an unknown environment and building a map of that environment using a simple circle detection algorithm. The map is updated by adding each new frame to the existing map, and the robot's position is tracked by updating its coordinates in each iteration of the loop. Finally, the map is displayed using the OpenCV library.

5. Robot Planning and Decision-Making:

Robot planning and decision-making are critical components of robotics that involve designing algorithms that enable robots to perform specific tasks. The following are some of the important algorithms used in robot planning and decision-making:

a. A* Algorithm: 
The A* algorithm is a popular algorithm used in robot path planning. It finds the shortest path between two points in a grid map. Here's an example implementation of the A* algorithm in Python:

python code

def astar(start, goal, grid):

    # define heuristic function

    def heuristic(a, b):

        return abs(b[0] - a[0]) + abs(b[1] - a[1])   

    # initialize open and closed sets

    open_set = [start]

    closed_set = []  

    # initialize g and f scores

    g_score = {start: 0}

    f_score = {start: heuristic(start, goal)}

    while open_set:

        # get the node with the lowest f score

        current = min(open_set, key=f_score.get)      

        # check if the goal is reached

        if current == goal:

            path = []

            while current in came_from:

                path.append(current)

                current = came_from[current]

            path.append(start)

            return path[::-1]

        # remove current from the open set and add to a closed set

        open_set.remove(current)

        closed_set.append(current)

        # Explore neighbours of the current node

        for neighbor in neighbors(current, grid):

            # skip neighbour if it is already in a closed set

            if neighbour in closed_set:

                continue      

            # calculate tentative g score

            tentative_g = g_score[current] + 1           

            # Add the neighbour to the open set if it is not already in

            if the neighbour is not in open_set:

                open_set.append(neighbor)

            elif tentative_g >= g_score[neighbor]:

                # skip neighbour if it already has a better g score

                continue        

            # Update g and f scores of neighbour

            came_from[neighbor] = current

            g_score[neighbor] = tentative_g

     f_score[neighbor] = g_score[neighbor] +                                                                           heuristic(neighbor, goal)

    # No path found

    return None

b. D* Algorithm: 

The D* algorithm is another path-planning algorithm used in robotics. It is similar to the A* algorithm but can handle dynamic environments. Here's an example implementation of the D* algorithm in Python:

sql code

def dstar(start, goal, grid):

    # define heuristic function

    def heuristic(a, b):

        return abs(b[0] - a[0]) + abs(b[1] - a[1])   

    # initialize open and closed sets

    open_set = [start]

    closed_set = []

    # initialize g and rhs scores

    g_score = {start: 0}

    rhs_score = {start: heuristic(start, goal)}   

    while open_set:

        # get the node with the lowest f score

        current = min(open_set, key=lambda x: (g_score[x] + rhs_score[x]))       

        # check if the goal is reached

        if current == goal:

            path = []

            while current in came_from:

                path.append(current)

                current = came_from[current]

            path.append(start)

            return path[::-1]       

        # remove current from the open set and add to a closed set

        open_set.remove(current)

        closed_set.append(current)

        # Explore neighbours of the current node

        for neighbor in neighbors(current, grid):

            # skip neighbour if it is already in a closed set

            if neighbour in closed_set:

                continue

            # calculate tentative g

tentative_g = g[current] + distance_between(current, neighbor)         

    # Add a new node to open the set if it's not there or if the new path is better

           if neighbor not in open_set or tentative_g < g[neighbor]:

                g[neighbor] = tentative_g

                f[neighbor] = g[neighbor] + heuristic(neighbor, goal)

                came_from[neighbor] = current          

                # Add the neighbour to the open set if it's not already there

                if the neighbour is not in open_set:

                    open_set.add(neighbor)                 

        # check if the goal has been reached

        if current == goal:

            path = []

            while current in came_from:

                path.append(current)

                current = came_from[current]

            path.reverse()

            return path, g[goal]     

    # goal not reachable

    return None, None

In this implementation, neighbours (current, grid) return a list of neighbours of the current node in the grid. distance_between(current, neighbour) calculates the distance between the current node and its neighbour. The Heuristic (neighbour, goal) calculates the heuristic value for the neighbour node, which is used to estimate the remaining cost to reach the goal. The g, f, and came_from dictionaries are used to keep track of the cost to reach each node, the total estimated cost to reach the goal through each node, and the parent node of each visited node, respectively. The open_set and closed_set sets are used to keep track of nodes that have been visited and are awaiting evaluation and nodes that have already been evaluated, respectively.

Overall, the D* algorithm is a useful path-planning algorithm for robotics because it can handle dynamic environments where the cost of moving between nodes can change over time.

6. Expert Systems:

Expert systems are computer programs that mimic the decision-making ability of a human expert in a specific domain. They work by applying a set of rules and knowledge to a specific problem, to provide a recommendation or solution.

Building a Rule-Based Expert System with PyKnow:

A Python package called PyKnow is used to create rule-based expert systems. Here's an example of how to use PyKnow to build a simple expert system for diagnosing a medical condition:

First, install the known library

diff code

!pip install pyknow

Next, define the knowledge classes and rules for the expert system:

python code

from pyknow import *

class Symptom(Fact):

    """Symptom fact"""

    pass

class Diagnosis(Fact):

    """Diagnosis fact"""

    pass

class MedicalExpertSystem(KnowledgeEngine):

    @Rule(Symptom('fever') & Symptom('cough'))

    def rule1(self):

        self.declare(Diagnosis('cold'))

Rule(Symptom('fever') & Symptom('shortness_of_breath'))

    def rule2(self):

        self.declare(Diagnosis('pneumonia'))

    @Rule(Symptom('fever') & Symptom('rash'))

    def rule3(self):

        self.declare(Diagnosis('measles'))

    @Rule(Symptom('fever'))

    def rule4(self):

        self.declare(Diagnosis('unknown'))

In this example, we define two fact classes (Symptom and Diagnosis) to represent the symptoms and diagnosis of the patient. We then define a MedicalExpertSystem class that inherits from Knowledge Engine and define several rules using the @Rule decorator. Each rule is defined in the form of a function, which takes as input a set of Symptom facts and declares a Diagnosis fact based on the presence of those symptoms.

Finally, we can use the expert system to diagnose a patient by creating instances of the Symptom class and running the expert system:

python code

expert = MedicalExpertSystem()

expert.reset()

expert.declare(Symptom('fever'))

expert.declare(Symptom('cough'))

expert.run()

print(expert.facts)

This will output:

CSS code

FactList([(Symptom(five...)])

FactList([(Symptom(five...), 1), (Diagnosis(cold), 1)])

In this example, we declare two Symptom facts (fever and cough) and run the expert system. The system applies the rules and declares a Diagnosis fact of a cold, based on the presence of those symptoms. The final output shows the list of facts that were used and generated during the inference process.

Note that this is a very simple example of a rule-based expert system, and real-world systems can be much more complex and sophisticated.

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