6.10 - Chaining Transformations (Summary)¶

The previous lesson demonstrated a robot arm with two linkages. This demonstrated how matrix transformations are chained together to create complex motion. The creation of the rendering transformations were created explicitly to make it clear how the transformations are created. Here is a summary of the transforms and the code that created them:

baseTransform =baseRotation
Eq1

forearmTransform =baseRotation
*translateToPin
*rotateForearm
Eq2

upperArmTransform =baseRotation
*translateToPin
*rotateForearm
*translateToForearmEnd
*rotateUpperarm
Eq3

// For rendering the base
matrix.multiplySeries(transform, projection, view, base_y_rotate);

// For rendering the forearm
matrix.multiplySeries(transform, projection, view, base_y_rotate,
                      forearm_translate, forearm_rotate);

// For rendering the upper arm
matrix.multiplySeries(transform, projection, view, base_y_rotate,
                      forearm_translate, forearm_rotate,
                      upperarm_translate, upperarm_rotate);

Reusing Transformations¶

Matrix multiplication is an expensive operation. For 4-by-4 matrices, multiplication requires 64 multiplies and 48 additions. We would like to avoid duplicate matrix multiplications whenever possible. For the robot arm, there is no need to repeat the same matrix multiplications over and over again. We can reuse the previous transform and simply post-multiply the additional transforms. The creation of each transform in the code could be done like this:

// For rendering the base
matrix.multiplySeries(transform, projection, view, base_y_rotate);

// For rendering the forearm
matrix.multiplySeries(transform, transform, forearm_translate, forearm_rotate);

// For rendering the upper arm
matrix.multiplySeries(transform, transform, upperarm_translate, upperarm_rotate);

Notice that in the 2^nd and 3^rd function calls the second parameter is the value of transform from the previous calculation. The following demo code contains theses simplified calculations.

Reusing transform calculations is important for complex renderings, but you should not collapse your transform calculations until you are completely clear on how to build complex chains of transformations. First, get a WebGL program to work correctly. You can always make efficiency optimizations later if slow rendering is an issue.

Show: Code Canvas Run Info

simple_transform_example_render4.js

/**
 * simple_transform_example_render4.js, By Wayne Brown, Spring 2016*
 */

/**
 * The MIT License (MIT)
 *
 * Copyright (c) 2015 C. Wayne Brown
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in all
 * copies or substantial portions of the Software.

* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 * SOFTWARE.
 */

"use strict";

//-------------------------------------------------------------------------
/**
 * Initialize and render a scene.
 * @param learn Learn_webgl An instance of Learn_webgl
 * @param vshaders_dictionary Object a set of vertex shaders
 * @param fshaders_dictionary Object a set of fragment shaders
 * @param models Object a set of models
 * @param controls Array a list of control id's
 * @constructor
 */
window.SceneSimpleExampleRender4 = function (learn, vshaders_dictionary,
                                         fshaders_dictionary, models, controls) {

// Private variables
  var self = this; // Store a local reference to the new object.

var out = learn.out;
  var events;
  var canvas;

var gl = null;
  var program = null;
  var render_models = {};

var matrix = new Learn_webgl_matrix();
  var transform = matrix.create();
  var projection = matrix.createOrthographic(-10, 10, -2, 18, -20, 20);
  var view = matrix.create();
  var base_y_rotate = matrix.create();
  var forearm_rotate = matrix.create();
  var forearm_translate = matrix.create();
  var upperarm_rotate = matrix.create();
  var upperarm_translate = matrix.create();

// Set the forearm_translate to a constant translation along the Y axis
  matrix.translate(forearm_translate, 0, 2, 0);
  matrix.translate(upperarm_translate, 0, 8, 0);

// We don't have a real view transform at this time, but we want to look
  // down on the model, so we will rotate everything by 10 degrees.
  matrix.rotate(view, 10, 1, 0, 0);

// Public variables that will be changed by event handlers or that
  // the event handlers need access to.
  self.canvas_id = learn.canvas_id;
  self.base_y_angle = 0.0;
  self.forearm_angle = 0.0;
  self.upperarm_angle = 0.0;
  self.animate_active = true;

//-----------------------------------------------------------------------
  // Public function to render the scene.
  self.render = function () {

// Clear the entire canvas window background with the clear color and
    // the depth buffer to perform hidden surface removal.
    gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);

// The base is being rotated by the animation callback so the rotation
    // about the y axis must be calculated on every frame.
    matrix.rotate(base_y_rotate, self.base_y_angle, 0, 1, 0);

// Combine the transforms into a single transformation
    matrix.multiplySeries(transform, projection, view, base_y_rotate);

// Draw the base model
    render_models.base.render(transform);

// Set the rotation matrix for the forearm; rotate about the Z axis
    matrix.rotate(forearm_rotate, self.forearm_angle, 0, 0, 1);

// Calculate the transform for the forearm
    matrix.multiplySeries(transform, transform, forearm_translate, forearm_rotate);

// Draw the forearm model
    render_models.forearm.render(transform);

// Set the rotation matrix for the upperarm; rotate about the Z axis
    matrix.rotate(upperarm_rotate, self.upperarm_angle, 0, 0, 1);

// Calculate the transform for the upperarm
    matrix.multiplySeries(transform, transform, upperarm_translate, upperarm_rotate);

// Draw the upperarm model
    render_models.upperarm.render(transform);
  };

//-----------------------------------------------------------------------
  // Public function to delete and reclaim all rendering objects.
  self.delete = function () {
    // Clean up shader programs
    gl.deleteShader(program.vShader);
    gl.deleteShader(program.fShader);
    gl.deleteProgram(program);
    program = null;

// Delete each model's buffer objects
    render_models.base.delete();
    render_models.forearm.delete();
    render_models.upperarm.delete();
    render_models = null;

// Disable any animation
    self.animate_active = false;

// Remove all event handlers
    events.removeAllEventHandlers();
    events = null;

// Release the GL graphics context
    gl = null;
  };

//-----------------------------------------------------------------------
  // Object constructor. One-time initialization of the scene.

// Get the rendering context for the canvas
  canvas = learn.getCanvas(self.canvas_id);
  if (canvas) {
    gl = learn.getWebglContext(canvas);
    if (!gl) {
      return;
    }
  }

// Enable hidden-surface removal
  gl.enable(gl.DEPTH_TEST);

// Set up the rendering shader program and make it the active shader program
  program = learn.createProgram(gl, vshaders_dictionary.shader05, fshaders_dictionary.shader05);
  gl.useProgram(program);

// Create the Buffer Objects needed for this model and copy
  // the model data to the GPU.
  render_models.base = new Learn_webgl_model_render_05(gl, program, models.Base, out);
  render_models.forearm = new Learn_webgl_model_render_05(gl, program, models.Forearm, out);
  render_models.upperarm = new Learn_webgl_model_render_05(gl, program, models.Upperarm, out);

// Set up callbacks for the user and timer events
  events = new SimpleTransformExampleEvents4(self, controls);
  events.animate();
};

A robot arm rendered by re-using transforms.

-90.0 +90.0 Forearm angle: 0.00 :
-90.0 +90.0 Upper arm angle: 0.00 :
Animate

Shader errors, Shader warnings, Javascript info

Open this webgl program in a new tab or window

Next Section - 6.11 - More Matrix Math Concepts